harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\langle \mu_R \rangle_\alpha^{\frac1q}\lesssim \langle\mu\rangle_\alpha^{\frac1q}\times \begin{cases}\, R^{\frac{n+1-2\alpha}{2q}}, n< \alpha \le n+1, \, R^{\frac{1-\alpha}{2q}}, 1<\alpha\le n, \, 1, 0< \alpha \le 1.\end{cases}
a_t :=\nabla f(x_t)+\nabla g(x_t)(\lambda_t+b_t) =\nabla f(\overline{x})+t\nabla^2f(\overline{x})\xi^* +(\nabla g(\overline{x})+t\nabla(\nabla g(\cdot))(\overline{x})\xi^*+o(t))(\overline{\lambda}+tv^*+b_t)+o(t) =t\nabla_{xx}^2L(\overline{x},\overline{\lambda})\xi^* +t\nabla g(\overline{x})v^*+o(t)=o(t)
L_{1}^{\left( 0 \right)}\, = \, - \frac{a}{2}A_{\mu}^{\left( 0 \right)}\, A^{\mu \left( 0 \right)}\, + \, \frac{b}{2}\left( F_{\mu \nu}^{\left( 0 \right)}\right)^{2}\, + \, \theta \, \epsilon^{\mu \nu \rho}\, \, A_{\mu}^{\left( 0 \right)}\, \partial_{\nu}\, A_{\rho}^{\left( 0 \right)}.
\Psi^\prime(r) = -\sum_{j=0}^2\alpha_j\int_{\partial^*\! E_j}\varphi^\prime\left(\frac{|x|}{r}\right) \frac{|x|}{r^2}\frac{(x\cdot\nu_{E_j}(x))^2}{|x|^2}\, d\mathcal{H}^{n-1}(x), r\in(0,d).
\begin{array}{c}{{G_{\rho \sigma \nu}^{a b c d}= \beta_{1}( G_{\rho \sigma \nu}^{( 1 )}+ G_{\rho \sigma \nu}^{( 2 )}) - \beta_{2}( G_{\rho \sigma \nu}^{( 1 )}+ G_{\rho \sigma \nu}^{( 3 )})}}\\{{+ \beta_{3}( G_{\rho \sigma \nu}^{( 3 )}- G_{\rho \sigma \nu}^{( 2 )}) = 0}}\\ \end{array}
\tilde{P}_{2sq_{n-n_0}-1}(\theta)=\sum_{x_1\in I_1\cup I_2}\tilde{P}_{2sq_{n-n_0}-1}(\theta_{x_1})\prod_{\substack{j\in I_1\cup I_2 j\neq x_1}}\frac{\sin\pi(\theta-\theta_j)}{\sin\pi(\theta_{x_1}-\theta_j)}.
\vec{p}_{0}=\frac{d_{1}(\vec{n}_{2}\times\vec{n}_{3})+d_{2}(\vec{n}_{3}\times\vec{n}_{1})+d_{3}(\vec{n}_{1}\times\vec{n}_{2})}{\vec{n}_{1}\cdot(\vec{n}_{2}\times\vec{n}_{3})}.
I_2= \sum_{j=1}^{i-1}x_j n_j \Delta x_j \bigg[\int_{x_{i-1/2}-x_j}^{x_{\alpha_{i-1,j}-1/2}}\beta(x_j,v)n(t,v)dv & + \sum_{k=\alpha_{i-1,j}}^I \beta_{j,k}n_k \Delta x_k+ \sum_{k=\alpha_{i-1,j}}^I \int_{\Lambda_k}(v-x_k)^2/2 (\beta(x_j,v)n(t,v))^{''}\bigg]dv+{\cal O}(\Delta x^3).
\Bigl|\frac{(t_i + (-1)^i r_1)(t_i + (-1)^i r_2)(t_i + (-1)^i r_3)}{\Upsilon_i t_i^2(t_1+t_2)}- \alpha_i \Bigr| \ll T^{\varepsilon}E_i^{-1/2}, i = 1, 2, \alpha_i \in \{\pm 1\}
\iota_\mathbf{n}(U_1 U_2) = \rho(U_1)\iota_\mathbf{n}(U_2) + \sum_\mathbf{m}\tbinom{\mathbf{n}+\mathbf{m}}{\mathbf{m}}\iota_{\mathbf{n}+\mathbf{m}}(U_1)\epsilon_\mathbf{m}(U_2),
\|I_2(D^{\beta_\perp}\sigma_\rho)\|_{L^2_{\gamma,\mu}(C(\Lambda[0,\frac{L}{2}],2\rho))}\le \|I_2(D^{\beta_\perp}\sigma_\rho)\|_{L^2_{\gamma,\mu}({\bf C}_0^{2\rho})}+ \|I_2(D^{\beta_\perp}\sigma_\rho)\|_{L^2_{\gamma,\mu}(\cup_{j=1}^{J-1}{\bf C}_j^{2\rho})}=:I+II.
\overline J_{\dot\alpha}^0 =-{i\over2}\overline\sigma^{0\dot\beta\beta}(\overline D_{\dot\alpha}R_{\beta\dot\beta}-2\varepsilon_{\dot\alpha\dot\beta}\overline D^{\dot\gamma}R_{\beta\dot\gamma})\bigr|_{\theta=\overline\theta=0}.
x_{k}(t)=F_{0}\omega\sum_{i=1}^{n-1}\frac{\phi_{i}(h)\phi_{i}(k)}{\omega^{2}-\omega_{i}^{2}}(\frac{sin\omega_{i}t}{\omega_{i}}-\frac{sin\omega t}{\omega})+\frac{F_{0}}{n}\frac{\omega}{\omega^{2}-1}(sint-\frac{sin\omega t}{\omega})
\bigl[ A^{b + \frac{\sigma}{2}}S(\,\cdot\,) \bigr]^{(j)}(t) = \bigl[ S( \,\cdot\, - \varepsilon) A^{b+ \frac{\sigma}{2}}S( \varepsilon ) \bigr]^{(j)}(t) = (-A)^j S(t-\varepsilon) A^{b + \frac{\sigma}{2}}S(\varepsilon) = (-1)^j A^{j + b + \frac{\sigma}{2}}S(t).
S_{q}={\frac{\pi}{3}}T_{H}( L - r_{+}) -{\frac{1}{1 2}}({\frac{r_{-}}{r_{+}}})^{2}\ln ({\frac{L - r_{-}}{r_{+}- r_{-}}}) +{\frac{1}{1 2}}\ln ({\frac{L - r_{+}}{r_{+}- r_{-}}}) +{\frac{1}{6}}\ln{\frac{r_{+}}{z_{0}}}~ ~ ,
X^2_0(t)= \sum_{k+\ell\leq s}\sum_{1\leq j\leq J}|x^{(k,\ell)}_j(t)|^2,\X^2_\pm(t)=\sum_{k+\ell\leq s}\sum_{i: \pm\lambda_i>0}|\xi^{(k,\ell)}_i(t)|^2.
{\mathrm R e}\partial_{p^{2}}\Gamma_{+ -}^{T}( p^{2}) \big |_{p^{2}= \kappa_{W}^{2}}= 1 \quad{\mathrm R e}\partial_{p^{2}}\Gamma_{Z Z}^{T}( p^{2}) \big |_{p^{2}= \kappa_{Z}^{2}}= 1 \quad{\mathrm R e}\partial_{p^{2}}\Gamma_{A A}^{T}( p^{2}) \big |_{p^{2}= \kappa_{A}^{2}}= 1
\hat{\varphi}(\omega,r){(g,h_1,h_2)}= \upsilon^{-1}\left( \varphi( \sigma^{(g,h_1,h_2)^{-1}}(\omega,r))|_{0,\dots,\kappa-1}\right) & = \upsilon^{-1}\left( T^{g^{-1}}(\rho(y))|_{0,\dots,\kappa-1}\right) & = \upsilon^{-1}\left( (\rho(\sigma^{g^{-1}}(y)))|_{0,\dots,\kappa-1}\right) & = \upsilon^{-1}\left( \upsilon( y(g)) \right) & = y(g)
|0 \rangle \rightarrow |0 \rangle_{\beta}= (1 +{\rm e}^{- \beta\epsilon})^{-1/2}\{|0 \rangle_a \otimes |0\rangle_b +{\rm e}^{- \beta \epsilon / 2}a^{\dagger}{\tilde a}|0 \rangle_a \otimes b^{\dagger}{\tilde b}|0 \rangle_b \}.
\lim_{r\rightarrow 0}\bigg(\int_{\partial U_{r}(z)}g_{X,\mathrm{hyp}}(z,\zeta)d_{\zeta}^{c}P_{X}(\zeta)-\int_{\partial U_{r}(z)}P_{X}(\zeta)d_{\zeta}^{c}g_{\mathrm{hyp}}(z,\zeta)\bigg)= P_{X}(z).
\Lambda_{{\rm b}\perp}=\tau\left(T_{\rm b}T_{\rm c}-T_{\rm{bc}}^2\right)-\left(t_{\rm{c4}}^2T_{\rm b}-2t_{\rm{c4}}t_{\rm{b4}}T_{\rm{bc}}+t_{\rm{b4}}^2T_{\rm c}\right) ,
Z_{\rm SG}\sim\sum_{n=0}^{\infty}\frac{1}{(n!)^2}\left(\frac{m^4}{2\lambda}\right)^{2n}\prod_{i}^{n}\int d^2 x_i d^2y_i \exp\Bigg(- \frac{\lambda}{m^2}\Big( \sum_{i < j}[\Delta (x_i - x_j) + \Delta (y_i - y_j)]- \sum_{i,j}\Delta(x_i - y_j) \Big)\Bigg).
\ell_{i,j}= (s_i)(s_{i+1}s_i)(s_{i+2}s_{i+1}s_i)\hdots(s_{j-1}\hdots s_i)(s_j\hdots s_i)= (s_i\hdots s_j)\ell_{i,j-1}=\ell_{i,j-1}(s_{j}\cdots s_{i}) &= (s_{j})(s_{j-1}s_{j})(s_{j-2}s_{j-1}s_{j})\hdots(s_{i+1}\hdots s_{j})(s_{i}\hdots s_{j}) = (s_{j}\hdots s_{i})\ell_{i+1,j}=\ell_{i+1,j}(s_{i}\hdots s_{j}).
\widehat{F}_{t, i}^{(u)}(\cdot) \, :=\, u \cdot \delta_{X_{t, i+1}^{(u)}}(\cdot) + (1-u) \cdot \frac{1}{n}\sum_{j=1}^{n}\delta_{X_{t, j}^{(u)}}(\cdot) \,, i \, =\, 1, \ldots , n-1 \,
m ( r ,{\kern 2 p t}t ) = \mu{\kern 1 p t}\frac{2 \Lambda / \pi}{r}{\kern 1 p t}\Big | \operatorname{s i n}\big ( \frac{\pi r}{2 \Lambda}\big ) \Big | \Big | \operatorname{s i n}\big ( \frac{\pi t}{2 T}\big ) \Big |
W \left[ C \right] = e^{- T E_{c l a s s}}\frac{\det \left( - \Delta_{F}- \frac{1}{4}R^{( 2 )}+ 1 \right)}{\det \left( - \Delta + 2 \right) \det^{\frac{1}{2}}\left( - \Delta + 4 - R^{( 2 )}\right) \det^{\frac{5}{2}}\left( - \Delta \right)}.
Z_f(s, \chi, D) = \int_{D}\chi(acf(x))|f(x)|^s|dx| = \int_{D_1}\chi(acf(\pi^{t_1}y_1, \cdots, \pi^{t_n}y_n))|f(\pi^{t_1}y_1, \cdots, \pi^{t_n}y_n)|^s|J(T_{t_1, \cdots, t_n})||dy_1\cdots dy_n| = q^{-(t_1+\cdots+t_n)}\int_{D_1}\chi(acf(\pi^{t_1}y_1, \cdots, \pi^{t_n}y_n))|f(\pi^{t_1}y_1, \cdots, \pi^{t_n}y_n)|^s|dy|,
\tilde{\cal H}= \frac{1}{2}\, \pi_{a}^{2}+ \frac{1}{2}\,{\sigma^{\prime}}_{a}^{2}- ( \sigma_{a}\pi^{a}) \theta + \frac{1}{2}\lambda \bigl ( \sigma_{a}^{2}- 1 \bigr ) + \frac{1}{2}\, \sigma_{a}^{2}\theta^{2}.
\Gamma_{n(\mu)}(x,s) = \begin{dcases}0, (u^2\mu-2w/\pi)^2-4\Delta /\varpi\notin F_0^{\times, 2}; \frac{\eta(-\nu)\bigl(|\nu|^{-s}+\eta(\Delta/\varpi) |\Delta/\varpi|^{-s}\nu^{s}\bigr)}{\bigl|(u^2\mu-2w/\pi)^2-4\Delta /\varpi \bigr|^{1/2}}, (u^2\mu-2w/\pi)^2-4\Delta /\varpi\in F_0^{\times, 2}. \end{dcases}
H_{n,m}(\mathbb F) =\sum_{\kappa_1+\cdots+\kappa_m=n}t^{-\sum_{i=1}^m(m-i+1)\cdot \kappa_{i}}=\frac{t^{-n^2-nm}\cdot P_{n,m}(t)}{|GL_n(\mathbb F)|}= &=t^{-mn}\prod_{j=1}^{n}\frac{1-t^{m+j-1}}{1-t^{j}}.
\frac{1}{2\pi\mathrm{i}}\oint_{|z|=r_{1}}\frac{f'(z)}{f(z)}\mathrm{d}z-\frac{1}{2\pi\mathrm{i}}\oint_{|z|=r_{1}}\frac{g'(z)}{g(z)}\mathrm{d}z=\sum_{|z_{i}|=r_{1}}\frac{1}{2\pi\mathrm{i}}\oint_{|z|=r_{1}}\frac{1}{z-z_{i}}\mathrm{d}z=\frac{1}{2}\#\{|z_{i}|=r_{1}\}.
\Psi ( x ) = \sum_{\sigma = 0 , \pm 1}\int{\frac{d^{3}p}{( 2 \pi )^{3}}}{\frac{1}{2 \omega_{\vec{p}}}}\Big [ u_{\sigma}( \vec{p}) a_{\sigma}( \vec{p}) e^{- i p \cdot x}+ v_{\sigma}( \vec{p}) b_{\sigma}^{\dagger}( \vec{p}) e^{i p \cdot x}\Bigr ] \quad ,
yz+xy\, M_1(1,x,y) \left(1+z\, M_1(1,x+y+xy\, M_1(1,x,y),z)\right) &= xy+yz\, M_1(1,y,z)\left(1+x\, M_1(1,x,y+z+yz\, M_1(1,y,z))\right)
F_{1}(\bar{v},\bar{w}) =\operatorname{div}\left( \mu\left( \rho_{0}\right) A_{\bar{v}}D_{A_{\bar{v}}}\left( \bar{w}\right) -\mu\left(\rho_{0}\right) D\left( \bar{w}\right) \right) , F_{2}(\bar{v},\nabla\bar{Q}) =\left( Id-A_{\bar{v}}^{T}\right)\nabla\bar{Q}.
\eta(r,z) = \begin{cases}\gamma|r|^{1-n}\left(-2[(1+|r|^{n-1})^\frac{1}{n-1}-1] \frac{z}{1+z^4}\, r,\ \arctan(z^2) (1+|r|^{n-1})^{\frac{2-n}{n-1}}|r|^{n-1}\right) z>0, 0 z\le 0\,.\end{cases}
R_4(t)=(1+t)\big(2+t+2t^2\big), R_5(t)= 2 (5,10,13,14,13,10,5)_{t}, R_6(t)= 10 (2,3,2)_{t}(7,7,10,13,10,13,10,7,7)_{t}, R_7(t)=30\big(196+\cdots+196 t^{15}\big).
\sum_{n=1}^{m-1}\frac{5m^8f(\frac{n}{m})}{\alpha(m,n)}\leq5m^8 \sum_{n=1}^{m-1}\frac{\frac{n^2}{m^2}}{n^4}\leq 5m^6\sum_{n=1}^{m-1}\frac{1}{n^2}\leq 5m^6\int_{1}^{m-1}\frac{1}{t^2}dt=5m^6\frac{m-2}{m-1}\leq 5m^6.
\mathbb{E}\sum_{j=0}^\infty\sum_{k=j+1}^\infty\left(P_{X,k}-Q_{X,k}\right) =\mathbb{P}(X=n)\sum_{j=0}^n\left(P_{n,n+1}-\sum_{k=j+1}^n\nu_kP_{n,n+1}\right) &=\mathbb{P}(X=n)\left(n+1-\mathbb{E}\nu\right)P_{n,n+1},
f^\star\sharp f(\xi) + h^\star\sharp h(\xi) = f(\xi)^*f(\xi) + h(\xi)^*h(\xi) + b(\xi) &= f(\xi)^*f(\xi) + 1 + g(\xi) + b(\xi) &= f(\xi)^*f(\xi) + 1 + C - f(\xi)^*f(\xi) + b(\xi) &= 1 + C + b(\xi) .
\left\| \partial_{t}^{\ell}\nabla_{x}^{k}\tilde{K}_{3}(t)g\right\|_{2}\le Ct^{1- \ell}(1+t)^{-n(\frac{1}{r_{1}}-\frac{1}{2})-(k-\tilde{k}_{3})}\| \nabla^{\tilde{k}_{3}}_{x}g \|_{r_{1}}& + C e^{-ct}t^{(1-\ell)-n(\frac{1}{r_{2}}-\frac{1}{2})-(k-\tilde{k}_{4})}\| \nabla^{\tilde{k}_{4}}_{x}g\|_{r_{2}},
\begin{cases}A^\top \nabla^{(1, 0)}\eta(Az_\star, y_\star) - y_\star^{(1)}\cdot e_d^{(1)}= \textbf{0}_d, \nabla^{(0, 1)}\eta(Az_\star, y_\star) - \tfrac{1}{p}\sum_{i=2}^{2T}(y_\star^{(i)})^p \cdot e_d^{(i)}- (z_\star^{(1)}- 2T + \tfrac{1}{p}) \cdot e_d^{(1)}= \textbf{0}_d. \end{cases}
\int (\Delta \phi)^2\dv = \int |\nabla^2 \phi|^2\dv + \int Ric(\nabla \phi,\nabla \phi)\dv &= \int |\nabla^2 \phi|^2\dv + (n-2) \int A(\nabla \phi,\nabla \phi)\dv + \int \sigma_1(A)|\nabla \phi|^2\dv,
\det(A_n) = \prod_{\substack{d_1, d_2 \mid n 1 \leqslant d_1 < d_2\leqslant n}}\left(\prod_{\substack{1 \leqslant i \leqslant \phi(d_1) 1 \leqslant j \leqslant \phi(d_2)}}(\omega_{d_2, j}- \omega_{d_1, i}) \right) = \prod_{\substack{d_1, d_2 \mid n 1 \leqslant d_1 < d_2 \leqslant n}}\mathcal{R}(\Phi_{d_2}, \Phi_{d_1})
r_1!r_2!\cdots r_t!\sum_{\tau\in T(r_1, \ldots, r_t)}a_{\tau(1)}a_{\tau(2)}\ldots a_{\tau(p)}=(r_1-1)!r_2!\cdots r_t!\sum_{\tau\in T(r_1, \ldots, r_t),\tau(1)=1}[[\cdots[[a_{\tau(1)},a_{\tau(2)}],a_{\tau(3)}],\ldots],a_{\tau(p)}].
\mathcal{D}\left((s_{k}')_{k=0}^{2m-3}, (1,0,\ldots,0) \right) \stackrel{a\to0+}{\longrightarrow}\mathcal{D}\left((-s^{(1)}_{k+1})_{k=0}^{2m-3}, (1,0,\ldots,0) \right) = (-1)^{m}\mathcal{E}\left((-s^{(1)}_{k+1})_{k=0}^{2m-3}, (1,0,\ldots,0) \right),
\mathbf{z}=k^{2}\int_{\Omega}g_{-k}\left( \left\Vert \mathbf{\cdot}-\mathbf{y}\right\Vert \right) \mathbf{v}\left( \mathbf{y}\right)d\mathbf{y}+\operatorname*{i}k\int_{\Gamma}g_{-k}\left( \left\Vert\mathbf{\cdot}-\mathbf{y}\right\Vert \right) T_{-k}\mathbf{v}^{\nabla}\left(\mathbf{y}\right) d\Gamma_{\mathbf{y}}.
M_\lambda(x) = (I - P_{\lambda,1}\Psi_\lambda(q))^{-1}\, ((\sqrt\lambda A + q I - (1/\sqrt\lambda) AQ)^{-1}+ P_{\lambda,1}M_f(x)) = M_\lambda(0) + (I - P_{\lambda,1}\Psi_\lambda(q))^{-1}P_{\lambda,1}M_f(x).
c_2 4^{-(n+1)}=g_{n+1}(t_{{n+1}})=M_n(g_n)(t_{n+1})\ge c_2 4^{-2n}\int_{0}^{t_{n+1}}g^2_n(\tau) dt\ge c_2 4^{-2n}\int_{t_n}^{t_{n+1}}c_2^2 4^{-2n}dt.
H(X_i|X^{i-1},\theta^i) \stackrel{(a)}\leq H(X_i|X_{i-1},\theta_i) \stackrel{(b)}= H(X_i|X_{i-1},\theta_i=0)\bar{\epsilon}\stackrel{(c)}= H(X_i|x_{i-1}=0,\theta_i=0)p(x_{i-1}=0|\theta_i=0)\bar{\epsilon}\stackrel{(d)}= H_b(p)p(x_{i-1}=0)\bar{\epsilon}
R(t;q) =(-qt;q)_\infty\sum_{k=0}^\infty\frac{(-1)^kq^{3k^2}t^{2k}}{(q^2;q^2)_k(-qt;q)_{2k}}&=(q^{1/2}t;q)_\infty\sum_{k=0}^\infty\frac{q^{k(3k-1)/4}t^k}{(q^{1/2};q^{1/2})_k(q^{1/2}t;q)_k}&=(q^{3/2}t;q)_\infty\sum_{k=0}^\infty\frac{q^{k(3k+1)/4}t^k}{(q^{1/2};q^{1/2})_k(q^{3/2}t;q)_k}.
\big\|P_s \big[f_*^{\frac1{p_s}}\big]\big\|_{L^{q_s}(\gamma)}/ \big(\int_{\mathbb{R}^n}f_*\, d\gamma \big)^{\frac1{p_s}}= e^{-\frac{n}{p_s}}(2\pi)^{-\frac{n}{q_s}}p_s^{\frac{n}2}e^{ns(2 - \frac1{q_s})}\Gamma(1-q_s)^{\frac{n}{q_s}}
\tilde{M}_\tau^\dagger \,\tilde{M}_\tau = \begin{pmatrix}X^2 0 0 0 \end{pmatrix}+ \tau \begin{pmatrix}0 e^{i \varphi}X D e^{-i \varphi}D^\dagger X 0 \end{pmatrix}+ \tau^2\: \begin{pmatrix}D D^\dagger+E 0 0 D^\dagger D \end{pmatrix}\:.
\mathcal{I}_{\mathrm{D}}^{\star}(\lambda)= \sup_{\pi \in \mathcal{G}_{\mathrm{D}, \lambda}}\int_{\mathcal{V}\times \mathcal{V}}\varphi_\lambda d \pi =\sup_{\pi \in \bar{\Gamma}}\int_{\mathcal{V}\times \mathcal{V}}\varphi_{\lambda}d \pi.
R ( \theta ) = \left( \begin{array}{c c c c}{1}&{0}&{0}&{0}\\{0}&{\frac{1}{( \theta + 1 )}}&{\frac{\theta}{( \theta + 1 )}}&{0}\\{0}&{\frac{\theta}{( \theta + 1 )}}&{\frac{1}{( \theta + 1 )}}&{0}\\{0}&{0}&{0}&{1}\\ \end{array}\right) .
{{\cal G}^{-1}}^{\alpha\beta}_{\mu\nu}\eta^{\rm Tr}_{\alpha\beta}=\left(k^{2}-\Lambda_{l}+\frac{1}{2r^{2}}\frac{N(N-1)(N-2)}{N+d-2}-\frac{N+d}{N+d-2}\Lambda\right)\eta^{\rm Tr}_{\mu\nu}.
f'(x_1^*,\ldots,x_L^*;0,\ldots,d_l,\ldots,0) =\nabla_{x_l}f(x_1^*,\ldots,x_L^*)^\top d_l &\le\|\nabla_{x_l}f(x_1^*,\ldots,x_L^*)\|_2 &=\left\|A_l^\top\left(b-\sum_{l\in[L]}A_lx_l^*\right)\right\|_2 &\le\|A_l\|_2\left\|b-\sum_{l\in[L]}A_lx_l^*\right\|_2 &\le\|A_l\|_2\left\|b-\sum_{l\in[L]}A_l\overline{x}_l\right\|_2,
\mathcal{E}_{m}^{\left( \alpha\right)}\left( \lambda\right) =2^{\alpha}\sum_{k=0}^{m}\genfrac{\{}{\}}{0pt}{}{m}{k}\binom{\alpha+k-1}{k}\frac{k!\left( -\lambda\right)^{k}}{\left(\lambda+1\right)^{\alpha+k}}.
\int_{0}^{1}\int\frac{1}{(k^{2}+m^{2}+2vp\cdot k+vp^{2})^{2}}dkdv=\int_{0}^{1}\int\frac{1}{(k^{\prime2}+m^{2}+v(1-v)p^{2})^{2}}dk^{\prime}dv
\sum_{\sigma}q^{\mathrm{inv}(\sigma)}=\prod_{s \in \lambda \cap S_\mu}\left( \sum_{a_s = 0}^{\rho(s,\mu,\lambda)-1}q^{a_s}\right)=\prod_{s \in \lambda \cap S_\mu}\frac{q^{\rho(s,\mu,\lambda)}- 1}{q-1}.
\det \begin{pmatrix}b_{1,\varphi}(x,\xi',\xi_d=\rho^+,\tau) b_{2,\varphi}(x,\xi',\xi_d=\rho^+,\tau) begin{align*}4pt]\partial_{\xi_d}b_{1,\varphi}(x,\xi',\xi_d=\rho^+,\tau) \partial_{\xi_d}b_{2,\varphi}(x,\xi',\xi_d=\rho^+,\tau)\end{pmatrix}\neq 0.
\delta^* = \alpha \frac{w_+^{1/2}(\rho)}{w_+^{1/2}(\lambda)}, \gamma^*= - \beta\frac{w_{-}^{1/2}(\rho)}{w_+^{1/2}(\lambda)}, \beta^* = - \frac{w_+^{1/2}(\lambda)}{w_{-}^{1/2}(\rho)}\gamma, \alpha^* = \frac{w_+^{1/2}(\lambda)}{w_+^{1/2}(\rho)}\delta.
|I|\lesssim \sum_{n=1}^N \gamma^{-n\alpha}\gamma^n h=h\sum_{n=1}^N\gamma^{n(1-\alpha)}=h\frac{\gamma^{1-\alpha}-\gamma^{N(1-\alpha)}}{1-\gamma^{1-\alpha}}=h\cdot O(h^{\alpha-1})=O(h^\alpha).
\bigcap_{i\in I}\bigcup_{j\in J_{i}}T_{i,j}=(\bigcup_{j\in J_{1}}T_{1,j})\cap(\bigcup_{j\in J_{2}}T_{2,j})=(\bigcup_{k\in K}T_{1,k})\cap(\bigcup_{l\in L}T_{2,l})=(\bigcup_{k\in K}C_{k})\cap(\bigcup_{l\in L}D_{l})
X(t) =X(0)+\int\limits_{0}^{t}b(X(u))du+\int\limits_{0}^{t}\sigma(X(t))dW(t)+\int\limits_{0}^{t}\int\limits_{E_{0}}\sigma^{0}(X({u-}),z)\widetilde{N}(du,dz) \ \ \ +\int\limits_{0}^{t}\int\limits_{E_{1}}\sigma^{1}(X({u-}),z)N(du,dz)+\int\limits_{0}^{t}\int\limits_{E_{2}}\sigma^{2}(X({u-}),z)N(du,dz),
\Psi_{0}^{(0)}(x;\omega ,N,\lambda_{1})=\psi_{0}^{( 0)}(x;\omega ,N)\left( \lambda_{1}+\int_{x_{0}}^{x}dt\frac{1}{\big( \psi_{0}^{(0)}(t;\omega ,N)\big)^{2}}\right) ,
P^*(y_2,\ldots,y_k;(O_{k,s}(M)L)^{\chi(y_2)},\ldots,(O_{k,s}(M)L)^{\chi(y_k)})\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\subset P_\textup{ord}(y_2,\ldots,y_k;(O_{k,s}(M)L)^{\chi(y_2)},\ldots,(O_{k,s}(M)L)^{\chi(y_k)})
S_\Gamma(g') =\left\{g\in\mathbb H^n: \delta_{d_K(g,g')^{-1}}(g'^{-1}\circ g)\in \Gamma\right\}, \Lambda_{\lambda}(g') =\bigg\{g\in\mathbb H^n: \delta_{d_K(g,g')^{-1}}(g'^{-1}\circ g)\in \Gamma,{|b(g)-b(g')|\over d_K(g,g')^Q}>\lambda\bigg\}.
\partial_\delta z_\delta= O\left(\frac{\delta^\frac{2-n}{2}}{(1+|\frac{x'}{\delta}|)^{n-3}}\right) \partial B_1(e_n)\cap B_1(0),\partial_\delta z_\delta= O\left(\delta^\frac{n-4}{2}\right) \partial B_1(e_n) \cap B_1^c(0).
\mathrm{div}^{\nabla'}(T_{0}) =0, \mathrm{div}^{\nabla'}(T_{r}) =\mathcal{A}_{\widehat{W}}\mathrm{div}^{\nabla'}(T_{r-1}) + \sum_{i=1}^{n}\epsilon_{i}(\overline{R}(\widehat{W},T_{r-1}Z_{i})Z_{i})',
(A-A^2 S')^k B^k (A-A^2 S')^k =(A^k-A^{k+1}S') B^k (A^k-A^{k+1}S') &=A^k B^k A^k-A^{k+1}S' B^k A^k-A^k B^k A^k B^k A^k S'+A^{k+1}(B^k A^k)^2{S'}^2 &=A^{k+1}-A^{k+2}S'=(A-A^2 S')^{k+1}.
\forall l\in I^{\varphi_\textup{KS}}(\bar x)\colon\xi_l:= \begin{cases}\mu_l/H_l(\bar x) l\in I^{0+}(\bar x), \nu_l/G_l(\bar x) l\in I^{+0}(\bar x), 0 l\in I^{00}(\bar x),\end{cases}
\mathcal{J}(\mu,\nu;\alpha):=\frac{1}{2}\mathbb{E}\Big\{\int_0^T\Big[Q\left(x_t^{\xi,\alpha}+l(\nu_t)\right)^2+R\left(\alpha_t+h(\mu_t)\right)^2\Big]dt+G\left(x_T^{\xi,\alpha}+g(\nu_T)\right)^2\Big\},
\int_0^{\pi}\dfrac{a}{c^{2s}}\,d\theta=\sum_{k=0}^s{2s+1\choose k}^2\rho^{4nk}I_0+2\sum_{l=1}^{s}\rho^{2nl}\sum_{i=0}^{s-l}{2s+1\choose i}{2s+1\choose i+l}\rho^{4ni}I_l,
\left( \begin{matrix}{{\Gamma^{\phi_{P}^{0}1 1}( p )}}&{{\Gamma^{\phi_{P}^{0}1 2}( p )}}\\{{\Gamma^{\phi_{P}^{0}2 1}( p )}}&{{\Gamma^{\phi_{P}^{0}2 2}( p )}}\\ \end{matrix}\right) = M_{P}^{- 1}\left( \begin{matrix}{{\Gamma_{R}^{\phi_{P}^{0}}( p )}}&{{0}}\\{{0}}&{{\Gamma_{R}^{\phi_{P}^{0}*}( p )}}\\ \end{matrix}\right) M_{P}^{- 1}
M_{i i}= \left( \begin{array}{c c}{{2 ( i - 2 )}}&{{0}}\\{{0}}&{{2 ( i - 1 )}}\\ \end{array}\right) + \left( \begin{array}{c c}{{\delta M_{\lambda \lambda}}}&{{\delta M_{\lambda \xi}}}\\{{\delta M_{\xi \lambda}}}&{{\delta M_{\xi \xi}}}\\ \end{array}\right) ,
\frac{d}{dt}\int_0^L\big(\frac{\rho u^2}{2}+\frac{\epsilon\chi_x^2}{2}+\frac{\rho(\chi^2-1)^2}{4\epsilon}\big)dx+\int_0^L\Big(\mu^2+\nu u_x^2+up_x(\rho)\Big)dxdt=0.
p \cdot \overline{C}_p = c \cdot d \cdot (c - 1) \cdot (d - 1) \cdot \left( \prod_{q \mid N_{\mathrm{sp}}}( 1- q^{-1}) \right) \cdot \left( \prod_{q \mid N_{\mathrm{ns}}}( 1 + q^{-1}) \right) \cdot \left( p - a_p(f) + \psi(p) \right) .
\epsilon n \leq |N^G_l(x)|= \sum_{i\in \Sigma}|N_l^G(x)\cap V_i|+\sum_{i\notin \Sigma}|N_l^G(x)\cap V_i| \leq |\Sigma|\frac{n}{t}+ (t-|\Sigma|)\frac{\epsilon}{2}\frac{n}{t}= |\Sigma|(1-\frac{\epsilon}{2})\frac{n}{t}+ \frac{\epsilon n}{2}.
\mathcal{C}(F,G)(x) := \textup{p.v.}\int_{\mathbb{R}}F(y) \frac{G(x)-G(y)}{(x-y)^2}dy & \,= \textup{p.v.}\int_{0}^{1}\! \int_{\mathbb{R}}F(y) \,G'((1-\kappa) x + \kappa y)\, \frac{dy}{x-y}\,d\kappa
S_1^{h_1}= \bigg( \sum_{2\le q \leq X}\Lambda(q) F_{\omega_{q}}(n)^2 \bigg)^{h_1}= \sum_{2\le q_1, \ldots, q_{h_1}\leq X}\prod_{i = 1}^{h_1}\Lambda(q_i) F_{\omega_{q_i}}(n)^2
L_{1}^{\ast}=\frac{\Gamma\left((m+1)/2\right)\sqrt{m}\left[\xi_{p}^{2}\left(1+\frac{\xi_{p}^{2}}{m}\right)^{-(m-1)/2}-\xi_{q}^{2}\left(1+\frac{\xi_{q}^{2}}{m}\right)^{-(m-1)/2}\right]}{\Gamma(m/2)(m-1)\sqrt{\pi}F_{Y}(\xi_{p},\xi_{q})},
F_{g,n}(z_{1},...,z_{n})=2^{2-2g-n}\sum_{d_{1}+...+d_{n}=3g-3+n}\prod_{i=1}^{n}\frac{(2d_{i}-1)!!}{z_{i}^{2d_{i}+1}}\langle\tau_{d_{1}}...\tau_{d_{n}}\rangle_{g}
\quad\;((nL,n\varphi)^{d+1})+\frac{1}{2}\nu(\Omega_\infty)(d+1)n^d(L^d)\ln(2r_n) &\geqslant((nL,n\widetilde \varphi_n)^{d+1})-\nu(\Omega_\infty)n^d(L^d)\Big((d+1)\ln(r_n)+\ln(n^d(L^d))\Big),
2 \kappa \int_{0}^{\infty}\frac{d p}{\pi p^{2}}\left( \frac{p \pi \coth p \pi}{( 1 + p^{2})}\Re \left[ \Phi_{p}^{L}( \tau ) \Phi_{- p}^{L}( \tau^{\prime}) \right] W_{\i j i^{\prime}j^{\prime}}^{L ( p )}( \chi ) - \Phi_{0}^{L}( \tau ) \Phi_{0}^{L}( \tau^{\prime}) W_{\i j i^{\prime}j^{\prime}}^{L ( 0 )}( \chi ) \right) ,
\begin{array}{lccc}\mbox{floor}\mbox{SU(1/2)}\mbox{SU(2)}\mbox{field}\noalign{\vskip3pt}\noalign{\hrule}\noalign{\vskip3pt}\mid \mbox{ground}> (0~1) (1) \lambda \mid \mbox{1st}> (-1~0) (0) \phi \end{array}
x^{t+1}-y^t=\ (1-\alpha_t)x^t+\alpha_tz^{t+1}-y^t = (1-\alpha_t)(x^t-y^t)+\alpha_tz^{t+1}-\alpha_t y^t \overset{\eqref{l2-e2}}{=}\ \alpha_t(1-\beta_t)(y^t-z^t)+\alpha_tz^{t+1}-\alpha_t y^t= \alpha_t\left(z^{t+1}-\beta_t y^t - (1-\beta_t)z^t\right).
S_{0}= \int d^{4}x \{p_{i}\dot{A}_{i}- \frac{p_{i}^{2}}{2}- \frac{1}{4}( \partial_{i}A_{j}- \partial_{j}A_{i})^{2}+ A_{0}\partial_{i}p_{i}+ \lambda \tilde{\partial}_{i}A_{i}\}
S \; = \; \int_{- \infty}^{\infty}\Bigl [ \;{\frac{1}{2}}\; ( 1 +{\frac{x^{2}}{\tau^{2}}}\; ( 1 - \varepsilon ) ) \left({\partial_{x}\phi}\right)^{2}+{\frac{\lambda}{4}}{\frac{1}{1 +{\frac{x^{2}}{\tau^{2}}}\; ( 1 - \varepsilon )}}\; \phi^{4}\; \Bigr ] \; d x ,
D^{(0)}(1||h) =\frac{1}{2\pi i}\oint_{|z|=1}\frac{1}{2}(\log{h}+\log{\bar{h}})^2\frac{dz}{z}&={\mathcal{K}}+\frac{1}{2\pi i}\oint_{|z|=1}\frac{1}{2}\big((\log{h})^2+(\log{\bar{h}})^2 \big)\frac{dz}{z}&={\mathcal{K}}+F(\boldsymbol{\xi})+\bar{F}(\bar{\boldsymbol{\xi}})
P^{-}={\frac{m^{2}}{4 \pi}}\int_{0}^{\infty}{\frac{d k}{k^{2}}}\left[ b^{\dag}( k ) b ( k ) + d^{\dag}( k ) d ( k ) \right] +{\frac{1}{2}}\int_{0}^{\infty}{\frac{d p}{p^{2}}}a^{\dag}( p ) a ( p ) \ .
\sum_{k=1}^{N}\mathbb{E}_{k-1}[\tilde{Y}_{k}(z_{1})\tilde{Y}_{k}(z_{2})] = \frac{1}{d_{N}^{2}}\frac{\partial^2}{\partial z_1 \partial z_2}s(z_1)s(z_2)+\tilde{C}_{N}(z_{1},z_{2})
S_{1 D}= - \int d^{5}\sigma \left( \sqrt{- G}\sqrt{1 + z_{1}+ \frac{z_{1}^{2}}{2}- z_{2}}+ \frac{\epsilon_{\mu \nu \lambda \sigma \tau}C^{\mu}\tilde{\cal H}^{\nu \lambda}\tilde{\cal H}^{\sigma \tau}}{8 ( 1 + C_{1}^{2})}\right) ,
\sum_{\varrho_f}\phi(\varrho_f)= \frac{k}{12}\cdot \frac{3}{\pi}\iint_{\mathcal F}\phi(z) \frac{dx dy}{y^2}+O_B(k \cdot h(k)^2) &+O_{A, B}\Big(k \cdot h(k) \log 1/h(k) \iint_{\mathcal F}|\Delta \phi(z) | \frac{dx \, dy}{y^2}\Big).
-( \lceil u,(\cdot)_T \rceil f)(x) =\int_{\mathbb{R}^2}\psi_T(x-y)(u(y)-u(x))f(y)dy, -(\lceil u,(\cdot)_T\rceil-\partial_1u \lceil x_1,(\cdot)_T\rceil )f(x) =\int_{\mathbb{R}^2}\psi_T(x-y)(u(y)-u(x)-(y-x)_1\partial_1u(x))f(y)dy
\frac1{2N+1}\sum_{i=-N}^{N+1-M}\Big(\sup_{0\le m\le M}\frac{1}{A_{m}}|\sum_{k =0}^{m}a_k v_{i+k}|\Big)^p \le \|\sup_{n\ge 0}\frac1{A_{n}}|\sum_{k=0}^{n}a_kf\circ \tau^k|\, \|_{L^p(\nu)}^p \le C^p\|f\|_{L^p(\nu)}^p= \frac{C^p}{2N+1}\sum_{i=-N}^N |v_i|^p\, .
H_{D}= \lambda ( \tau ) H ( \tau ) - \vec{\lambda}( \tau ) \vec{H}_{p}( \tau ) + \int{d^{3}\sigma \Big [ \mu_{\tau}( \tau , \vec{\sigma}) \pi^{\tau}( \tau , \vec{\sigma}) - A_{\tau}( \tau , \vec{\sigma}) \Gamma ( \tau , \vec{\sigma}) \Big ]},
W_2(k_1)=\frac{1}{2\pi R_c}\left[\int_0^{\pi R_c}dx\,<\frac{1}{N}{\rm Tr}\,\left[\,\Omega^{(k_1)}(x)\,\right]\frac{1}{N}{\rm Tr}\,\left[\,\Omega^{(-k_1)}(0)\,\right]>+\,k_1\to-k_1\right].
{\cal V}_{1,1}'^{(i)}\geq 0, \;\;{\cal V}_{2,2}'^{(i)}\geq 0, \;\;{\cal V}_{1,1}'^{(i)}\geq |{\cal V}_{1,2}'^{(i)}|, \;\;{\cal V}_{2,2}'^{(i)}\geq |{\cal V}_{2,1}'^{(i)}|.
\|(I-P_{M+N})x-x_n\| \leq C_1 D_p(x_n,(I-(P_{M+N}))x)^{1/\rho}= C_1 D_{p^*}(\Pi^{p^*}_{M\perp\cap N^\perp}y_0,y_n)^{1/\rho}\leq C_1C_2 (q')^{n/\rho}=: C q^{n}
\frac{d^a}{dX^a}\sum_{k=0}^{n-1}T_{\o_k}(1,R(X))X^k\Big|_{X=\l}=\frac{d^a}{dX^a}\sum_{k=0}^{n-1}T_{\o'_k}(1,R(X))X^k\Big|_{X=\l}
H_{n c}\; = \; \left( \nu_{n c}\: M + \overline{{{\nu_{n c}}}}\: \overline{{{M}}}+ L + \overline{{{L}}}\right) \: T_{[ 0 ]}^{( - 1 )}\: T_{[ 0 ]}^{( 0 )}\: \overline{{{T_{[ 0 ]}^{( - 1 )}}}}\; ,
\partial_{t}^{2}\mathcal{U}(x,t)=L_{t}^{\alpha}\partial_{x}\mathcal{E}_{x}^{\beta}\mathcal{U}(x,t)+u_{0}(x)\delta^{\prime}(t)+v_{0}(x)\delta(t),\;\;\;\;\mathcal{K}^{\prime}(\mathbb{R}^{2}).
\begin{aligned}\Theta_j = -\mu_j^{n-2s}\left(H(x, q_j) - \Phi_j^*(x, t)\right) & + \sum_{i\neq j}\left[(\mu_j\mu_i^{-1})^{\frac{n-2s}{2}}U(y_i)-(\mu_j\mu_i)^{\frac{n-2s}{2}}\left(H(x, q_i) - \Phi_i^*(x, t)\right)\right].\end{aligned}
\overset{\longleftarrow}{\prod_{\mu\in \mathbb{Q}_{> 0}}}A_{\mu}^{\theta}\star A^{\sigma, \theta}= \overset{\longleftarrow}{\prod_{\mu \in \mathbb{Q}_{> 0}}}A_{\mu}^{\theta^{\prime}}\star A^{\sigma, \theta^{\prime}}
\sum_{i=1}^{M}\lambda_i\mu_{i:M}- \left\lvert\sum_{i=1}^{M}\lambda_i\mu\right\lvert \le \left\lvert\sum_{i=1}^{M}\lambda_i\mu_{i:M}\right\lvert - \left\lvert\sum_{i=1}^{M}\lambda_i\mu\right\lvert & \le \left\lvert\sum_{i=1}^{M}\lambda_i(\mu_{i:M}-\mu)\right\lvert \le \sigma\sqrt{M\sum_{i=1}^{M}(\lambda_i-\bar{\lambda})^2}.
\mu\left\{x \in X \mid |A^{}_{N}f(x)| \geq \frac{2}{N^\gamma}\right\}\leq \mu\left\{x \in X \mid \left\lvert A^{}_{N}f^{N,\gamma}(x)\right\rvert \geq \frac{1}{N^\gamma}\right\}\leq C_{}\cdot N^{2\gamma}\left\|A^{}_{N}f^{N,\gamma}\right\|_{L^{2}(X,\mu)}^{2}.
S_{f , i}= \lim_{\tau \rightarrow - \infty , \tau^{\prime}\rightarrow + \infty}( - \sqrt{2}e^{\tau^{\prime}}\stackrel{\leftrightarrow}{\frac{\partial}{\partial \tau^{\prime}}}) ( \sqrt{2}e^{- \tau}\stackrel{\leftrightarrow}{\frac{\partial}{\partial \tau}}) G ( \tau^{\prime}, \tau )