harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\theta:= \sum^3_{a=1}\eta_a(\xi)\theta_a, \phi^\perp:=\sum^3_{a=1}\eta_a(\xi)\phi_a, \xi^\perp:=\sum^3_{a=1}\eta_a(\xi)\xi_a, \eta^\perp:=\sum^3_{a=1}\eta_a(\xi)\eta_a.
\left( \Psi \left| D^{3}\right| \Psi \right) = \left( \Psi \left| - D^{\mu}D_{\mu}\gamma^{\rho}D_{\rho}\right| \Psi \right) + \frac{1}{2}\left( \Psi \left| \gamma^{\mu \nu}F_{\mu \nu}\gamma^{\rho}D_{\rho}\right| \Psi \right) .
|I_1|=\left|h(n_1,E_1)\frac{(1-x)^\frac{1}{2}}{n_1-1}\frac{(1-x)^\frac{1}{2}}{n_2-1}\frac{n_2-n_1}{(1-x)^\frac{1}{2}}\right|\le C\frac{|n_1-n_2|}{(1-x)^\frac{1}{2}}.
u_\epsilon(X_k,Y_{k},t_{k}) =u_\epsilon(X_k,Y_{k-1}+{\epsilon^2}X_{k}/2,t_{k}) &\leq \inf_{\tilde X\in{B_\epsilon(X_{k-1})}}u_\epsilon(\tilde X,Y_{k-1}+{\epsilon^2}\tilde X/2,t_k)+\eta 2^{-k},
x_1= e_{23}+e_{56},~ y_1 = e_{32}+e_{65},~ h_1=[x_1,y_1] & x_2=e_{12}+e_{34}+e_{45}+e_{67},~ y_2=e_{21}+2e_{43}+2e_{54}+e_{76},~ h_2=[x_2,y_2].
f(A) \le f(2^K) + \sum_{\substack{p>2 p\in A}}f(p) + (2-2^{1-K})\sum_{\substack{2<p\le 23 p\notin A}}b_pf(p) + 2\sum_{\substack{p> 23 p\notin A}}b_p f(p) & \le f(2^K) + (2-2^{1-K})\sum_{2<p\le 23}b_pf(p) + 2\sum_{p> 23}b_p f(p) & =: \f(2^K) + (2-2^{1-K})C_1 + 2C_2.
K_\theta(y|y^\prime) = W_Y(y|y^\prime) T(y|y^\prime) \Bigg( \sum_x \!\bigg( \!\frac{W_{X|X',Y',Y}(x|x^\prime,y^\prime,y)}{T(y|y^\prime)}\!\bigg)^{1+\theta}\!\Bigg)^{\frac{1}{1+\theta}}\to W_Y(y|y^\prime) T(y|y^\prime)
Q_{\lambda, q}(\Omega) f^{q-1}(\xi)=\int_\Omega\frac{f(\eta)}{|\eta^{-1}\xi|^{Q-\alpha}}d\eta+ \lambda \int_\Omega\frac{f(\eta)}{|\eta^{-1}\xi|^{Q-\alpha-1}}d\eta,\xi\in \overline \Omega
\begin{gathered}\int_0^\infty C_2(u) \int_x^\infty \rho^{1-2\alpha-\nu}({x^2}/{\rho^2}; q^2)_{-\alpha}J_\nu (u\rho; q^2) \,d_q \rho\, d_q u = \int_x^\infty g_2 (\rho) \rho^{-2\alpha-\nu+1}({x^2}/{\rho^2}; q^2)_{-\alpha}\,d_q\rho. \end{gathered}
\begin{array}{l}H(y)=H_{b_n b_{n-1}}\dots H_{b_2 b_{1}},H(b_1)=\textrm{id}, \hat{H}(y)=H(y^{-1})^{\ast}=\hat{H}_{b_n b_{n-1}}\dots \hat{H}_{b_2 b_{1}}.\end{array}
\left( \lambda \begin{bmatrix}0 D(\lambda) \end{bmatrix}+B \right)\left(L_\epsilon(\lambda^\ell)\otimes I_n\right)+\left(L_\eta(\lambda^\ell)^T \otimes I_m\right)\left( \lambda \begin{bmatrix}0 -D(\lambda) \end{bmatrix}+C\right)
[{\cal T}_\alpha,{\cal T}^\dagger_\beta]=\bar{U}^\gamma_{\alpha\beta}{\cal T}_\gamma +{\cal T}^\dagger_\gamma\bar{U}^{\dagger\gamma}_{\beta\alpha}+[\Omega,\bar{U}^\gamma_{\alpha\beta}]\bar{C}_\gamma(-1)^{\varepsilon_\alpha+\varepsilon_\beta}+\bar{C}^\dagger_\gamma[\bar{U}^{\dagger\gamma}_{\beta\alpha},\Omega](-1)^{\varepsilon_\alpha+\varepsilon_\beta},
u(x) \geq M\int_{\Omega}{u(z) f_0(z)\, dz}=\frac{a_{N,s}}{2}\iint_Q\frac{(u(x)-u(y))(w(x)-w(y))}{|x-y|^{N+2s}}\,dxdy &= M\int_{\Omega}{w(z) f(z)\, dz}\geq C_0\int_{\Omega}{f(z)\overline{\xi_0}(z) \, dz}\geq \lambda w(x),
\mathrm{Reg}_{0}(X/k) = \deg_{X/k}: \mathrm{CH}_{0}(X) \otimes \Lambda \to \Lambda \mathrm{Reg}_{i}(X/k) = 0 : \mathrm{CH}_{0}(X,i) \otimes \Lambda \to 0 \ \ \ (if \i>0)
abc \leq \frac{a^p}{p}+ \frac{b^{p'}c^{p'}}{p'}\leq \frac{a^p}{p}+ \frac{1}{p'}\left(\frac{(b^{p'})^{q/p'}}{q/p'}+ \frac{(c^{p'})^{r/p'}}{r/p'}\right) = \frac{a^p}{p}+\frac{b^q}{q}+\frac{c^r}{r},
M ( p ) = \frac{e^{2}}{\pi}m \left[ \frac 3 4 \ln \left( 1 + \frac{L^{\prime 2}}{m^{2}}\right) + \frac{L^{\prime}}{2 m}\arctan \frac m{L^{\prime}}- \frac 1 8 \frac{L^{\prime 2}/ m^{2}}{1 + L^{\prime 2}/ m^{2}}\right] .
\mathcal{T}^s_{Tpr}:= \left\{f \ \vert \ \Vert f \Vert_{\mathcal{T}^s_{Tpr}}= \left\Vert (2^{qs}\Vert |\!|\!| \Delta_q f |\!|\!| \Vert_{L^{p}_T L^r_x})_{q \ge -1}\right\Vert_{\ell^{1}}<\infty \right\}.
\frac{\omega^2}{r^2}\,\frac{\partial z}{\partial \phi}(\phi,\omega,r;t) = \frac{\partial}{\partial s_1}F(0, p(\phi+\omega t))\cdot p'(\phi) + \frac{\partial}{\partial s_2}F(0,p(\phi+\omega t))\cdot p'(\phi+ \omega t).
\alpha_{n-j}(F) \geq{}\alpha_{n-j}(F_1/T_1) + \alpha_{n-j}(F_2/T_2) -{}\sum_{l=1}^{j-1}\frac{(-1)^{l-1}}{2l!}(a_2^l\alpha_{n-j+l}(F_1/T_1) + a_1^l\alpha_{n-j+l}(F_2/T_2)).
H_{\rm vis,\pm}^2=\frac{\rho_{\rm vis}(\rho_{\rm vis}+2\Lambda_2^\pm)}{36M^6(1+4\alpha\Lambda_b/3M^5)}=\frac{\pm \rho_{\rm vis}}{3M_{P}^2\sqrt{1+4\alpha\Lambda_b/3M^5}}\left[1+\frac{\rho_{\rm vis}}{2\Lambda_2^\pm}\right]~~.
H = ( a , a , 1 / 2 , - x ) ={\frac{\pi^{{\frac{- 1}{2}}}}{2}}\Gamma ( a +{\frac{1}{2}}) \Gamma ( 1 - a ) ( 1 + x )^{- a}\left( P_{2 a - 1}[{\frac{x^{\frac{1}{2}}}{\sqrt{( 1 + x )}}}] + P_{2 a - 1}[{\frac{- x^{\frac{1}{2}}}{\sqrt{( 1 + x )}}}] \right)
\mathbf{T}_N = \begin{bmatrix}\mathbf{1}_{N_s}\mathbf{0}\dots \mathbf{0}\mathbf{0}\mathbf{1}_{N_s}\dots \mathbf{0}\vdots \vdots \ddots \vdots \mathbf{0}\mathbf{0}\dots \mathbf{1}_{N_s}\end{bmatrix},
\phi_{3,i}= -\frac{n-2}{2}u_{a_{i}}\frac{\lambda_{i}\gamma_{n}\nabla_{a_{i}}G^{\frac{2}{2-n}}_{a_{i}}}{1+\lambda_{i}^{2}\gamma_{n}G^{\frac{2}{2-n}}_{a_{i}}}\varphi_{i}+\frac{\nabla_{a_{i}}u_{a_{i}}}{u_{a_{i}}\lambda_{i}}\varphi_{i}.
c_{\ell,0}\|\phi_{\ell,\phi,\beta}\|^{-2}_{L^2_{\rho_\beta}}\langle \varepsilon,\phi_{\ell, b, \beta}\rangle_{L^2_{\rho_\beta}}=- \|\phi_{0,\phi,\beta}\|^{-2}_{L^2_{\rho_\beta}}\langle \varepsilon,\phi_{0, b, \beta}\rangle_{L^2_{\rho_\beta}}
\widetilde\Delta^{(m)}\partial_{z_j}= \partial_{z_{j-1}}- 2 \partial_{z_j}+ \partial_{z_{j+1}}\,, \;\; 1 \le j \le m-2 \,, \widetilde\Delta^{(m)}\partial_{z_{m-1}}= \partial_{z_{m-2}}- \partial_{z_{m-1}}
U_\theta(f)U_\theta(g) = \int_{\mathbb{R}^d}f(t)U_\theta(t)dt\int_{\mathbb{R}^d}g(s)U_\theta(s)ds =\int_{\mathbb{R}^d}\int_{\mathbb{R}^d}f(t)g(s)e^{\frac{i}{2}(s,\theta t)}U_\theta(s+t)dtds =\int_{\mathbb{R}^d}\left(\int_{\mathbb{R}^d}f(t)g(s-t)e^{\frac{i}{2}(s,\theta t)}dt\right)U_\theta(s)ds.
Y^{(d)}(\Theta,x)=\partial_{\Theta_2}^d\,Y(\Upsilon_{\!(\Theta_1+\Theta_2)}x) &=\partial_{\Theta_2}^d\,\left[T(\Upsilon_{\!\Theta_1}^{-1})\,Y(\Upsilon_{\Theta_2}x)\right]=T(\Upsilon_{\!\Theta_1}^{-1})\,Y^{(d)}(\Theta_2,x)
\int_{0}^{1}d r_{0}f_{1}( r_{0}) = - \frac{\zeta ( 3 )}{1 2 8 \pi^{4}}- \frac{4 5 \zeta ( 3 )}{5 1 2 \pi^{6}}+ \frac{1 5 \zeta ( 5 )}{1 2 8 \pi^{6}}
\begin{array}{l}{{R ( \theta - \theta^{\prime}) K_{1}^{+}( \theta^{\prime})^{t_{1}}M_{1}^{- 1}R ( - \theta - \theta^{\prime}- 2 \eta ) K_{1}^{+}( \theta )^{t_{1}}M_{2}=}}\\{{K_{1}^{+}( \theta )^{t_{1}}M_{2}R ( - \theta - \theta^{\prime}- 2 \eta ) M_{1}^{- 1}K_{1}^{+}( \theta^{\prime})^{t_{1}}R ( - \theta - \theta^{\prime})}}\\ \end{array}
\mathcal{B}^+_{III}= \left\{\begin{aligned}\left(\tau,v_{s+k+1},\omega_{k+1}\right)\in\mathcal{A}^+\backslash\mathcal{B}^+_{II}\; \textnormal{such that}& \inf_{i\in\left\{1,\dots,s,s+1,\dots,s+k\right\}\backslash \left\{i_{k+1}\right\}}\left|v_{s+k+1}^* - v_i^\prime \right| \leq \eta\end{aligned}\right\}
\Xi ( n ) = C_{p}n^{k}\exp \left\{\frac{1 + p}{p}[ A \Gamma ( 1 + p ) \zeta_{R}( 1 + p ) ]^{\frac 1{1 + p}}n^{\frac{p}{1 + p}}\right\}\left( 1 +{\cal O}( n^{- k_{1}}) \right) \, \mathrm{,}
\begin{array}{rcl}a + b := \varphi \big( (a / \alpha) \oplus (b/\alpha), \, \psi_2 (a) +_o \psi_2 (b) \big) - b := \varphi \big( \ominus (b/\alpha), \, -_o \, \psi_2 (b) \big). \end{array}
T_{123}(p^2,m_1^2,m_2^2,m_3^2) = - \frac{2^4}{(2 \pi \mu^2)^{-2}}\mathcal{I}= - \frac{2^4}{(2 \pi \mu^2)^{-2}}\int_0^\infty dx~x~ J_0( \sqrt{- p^2}x) K_0(m_1 x) K_0(m_2 x) K_0(m_3 x)
\widehat R_T^{bp}={\cal T}(\hat{r}_{-(T-1),T}^{bp},\ldots,\hat{r}_{(T-1),T}^{bp}) \widehat R_T^{up}={\cal T}(\hat{r}_{-(T-1),T}^{up},\ldots,\hat{r}_{(T-1),T}^{up}).
Q_i := \exp 2\pi i \tau_i, \ \ \ \ \ \ \ \ \ \Q_{i,(a_1,\ldots,a_l)}:= \exp 2\pi i \tau_{i,(a_1,\ldots,a_l)}Q_{(i,j)}:= \exp 2\pi i \sigma_{(i,j)}, \ \ \Q_{(i,j),(a_1,\ldots,a_l)}:= \exp 2\pi i \sigma_{(i,j),(a_1,\ldots,a_l)}.
\begin{aligned}\|f''\|_1 \leq \limsup_{n\rightarrow \infty}|f(n+1)-f(n)| +\limsup_{n\rightarrow \infty}|f(-n-1)-f(-n)| + 2\sum_{n\in \partial_l S_-f}f(n)-f(n-1) + 2\sum_{n\in \partial_r S_-f}f(n)-f(n+1).\end{aligned}
\frac{1}{\lambda_{0}}= \frac{\sigma_{0}^{D - 2}}{\lambda_{r}}+ \int_{- \infty}^{\infty}\frac{d \omega}{2 \pi}\, \frac{1}{( 4 \pi )^{\frac{D - 1}{2}}}\left\{\omega^{2}+ ( D - 2 ) \sigma_{0}^{2}\right\}( \omega^{2}+ \sigma_{0}^{2})^{\frac{D - 5}{2}}\Gamma ( - \frac{D - 3}{2}) \mathrm{t r}\mathrm{\boldmath ~ 1 ~}.
\alpha_{ij}=[\begin{matrix}1&\alpha_{1}&0&0&0\\ \alpha_{-1}&1&\alpha_{1}&0&0\\ 0&\alpha_{-1}&1&\alpha_{1}&0\\ 0&0&\alpha_{-1}&1&\alpha_{1}\\ 0&0&0&\alpha_{-1}&1\end{matrix}]
\langle \phi | x^{m l}\rangle = N^{1 / 2}( n ) ( \beta / 2 \pi )^{n / 4}\int_{I_{n}}d^{n}\rho ~ ( 1 - \beta \rho^{2}/ 2 )^{1 / 2 + \sqrt{1 / 4 + n / 8}}e^{i x \cdot \rho / \hbar}\phi ( \rho )
\begin{array}{ll}\displaystyle\int_{B_{R}(0)}\frac{u^{q}(x\cdot \nabla u)}{|x|^{b(q+1)}}dx=\frac{1}{q+1}\int_{B_{R}(0)}\frac{x\cdot \nabla u^{q+1}}{|x|^{b(q+1)}}dx begin{align*}3mm] =\displaystyle\frac{R}{q+1}\int_{\partial B_{R}(0)}\frac{u^{q+1}}{|x|^{b(q+1)}}ds-\frac{n-b(q+1)}{q+1}\int_{B_R(0)}\frac{u^{q+1}}{|x|^{b(q+1)}}dx.\end{array}
G(v,Q) :=\tfrac{p+2}{2}\mu v\int_0^1\bigl[|Q+\theta v|^p-|Q|^p\bigr]\,d\theta + \tfrac{p}{2}\mu\bar v\int_0^1 \bigl[|Q+\theta v|^{p-2}(Q+\theta v)^2-|Q|^{p-2}Q^2\bigr]\,d\theta
Z \left(\frac{1}{\rho_V}, \rho_R \right) \rho_V^K = \sum\limits_{(k,l) \in \chi}\rho_V^{K-k}\frac{\rho_R^{l}}{l!}= \sum_{l=0}^K \frac{\rho_R^{l}}{l!}\sum_{k=0}^{K-l}\rho_V^{K-k}&= \sum_{l=0}^K \frac{\rho_R^{l}}{l!}\sum_{k'=0}^{K-l}\rho_V^{k'+l}=\sum\limits_{(k',l) \in \chi}\frac{\rho_R^{l}}{l!}\rho_V^{k'+l}&= Z(\rho_V, \rho_R\rho_V).
\sum_{n=-\infty}^{\infty}\omega^{4n}q^{2n(2n-1)}\left[\varphi_{k}\left(\omega q^{2n}\right)\varphi_{\ell}\left(\omega q^{2n}\right)+\omega^{2}q^{4n}\varphi_{k}\left(-\omega^{-1}q^{-2n}\right)\varphi_{\ell}\left(-\omega^{-1}q^{-2n}\right)\right] \hskip-24pt=\|\varphi\left(\omega\right)\!\|^{2}\delta_{k,\ell}
\begin{aligned}\int_{0}^{\pi}[r_{n}^{2}\omega(r_{n},\theta)^{2}+2r_{n}|\omega(r_{n},\theta)\omega_{\theta}(r_{n},\theta)|]d\theta\leq\frac{1}{\log 2}\int_{r>2^{n},0<\theta<\pi}(2\omega^{2}+|\nabla\omega|^{2})dxdy.\end{aligned}
V_{n,k,w}= \left( \frac{1}{n}\right)^{\frac{k}{2}}\sum_{\emptyset \subsetneq E_{f}\subsetneq E_{w}}\prod_{e \in E_{f}}\sigma_{e}(\frac{2\beta}{\sqrt{n}}) \prod_{e \in E_{w}\backslash E_{f}}B_{e}
\begin{aligned}\Bigl| \int \zeta g_{\sigma_n}\big ( \nabla_{g_{\sigma_n}}u_n, \overline{\nabla_{g_{\sigma_n}}u_n)}\big) - \bigl(\tau_n \zeta + \frac{\Delta_{g_{\sigma_n}}(\zeta)}2 \bigr) |u_n|^2\, dv_{g_{\sigma_n}}\Bigr| \leq \Vert u_n \Vert_{L^2(B_2)}\Vert f_n \Vert_{L^2(B_2)}= o(|\tau_n|^{1/2}),{n\rightarrow + \infty}.\end{aligned}
s_{m}\pi_{n}^{a}=-\delta_{mn}\lambda^{a}-\frac{1}{2}f_{bc}^{a}\eta_{m}^{c}\pi_{n}^{b}+\frac{1}{12}\varepsilon_{npr}f_{bc}^{a}f_{de}^{c}\eta_{m}^{e}\eta_{p}^{d}\eta_{r}^{b}\,, s_{m}p_{n}^{a}=-\delta_{mn}l^{a}\,,
\|{h}-{h_0}\|_{L^1(D_{\frac14})}=\|\widetilde{h}-\widetilde{h}_0\|_{L^1(D_{1})}&\leq C\|\tau(v)\|_{L^p(D_{1})}\leq C\|\tau(u)\|_{L^p(D_{1})}\leq C\|\tau(u)\|_{L^p(D_{1})}^{3^{1-m}}.
\int_{\Sigma}\int_{\Sigma}\frac{(\phi(x) - \phi(y) ) \phi(x)}{|x-y|^{N+s}}d x d y =\int_{\Sigma}\int_{\Sigma}\frac{(w(x) - w(y) ) w(x) \psi(x)^2}{|x-y|^{N+s}}d x d y & +\int_{\Sigma}\int_{\Sigma}\frac{(\psi(x) - \psi(y) ) w(x) w(y) \psi(x)}{|x-y|^{N+s}}d x d y .
VII\leq C\sum\limits_{k=1}^{\infty}2^{-2k}\Big(\|\vec{b}\|_{*}M_{\alpha,r,w}f(x)+\|\vec{b}\|_{*}M_{\alpha,1}f(x)\Big) &\leq C\|\vec{b}\|_{*}\Big(M_{\alpha,r,w}f(x)+M_{\alpha,1}f(x)\Big)
S_{n}:= \left\{s \mid s: [n] \times K \times \mathbb{Q}^{[0,1]}_n \to \{-1, 1\}\right\}, U_{n}:= \left\{u \mid u: [n] \times \mathbb{Q}_n^{[0,1]}\to \{-1, 1\}\right\}, \textrm{and}Q_{n}:= S_{n}\times S_n \times U_{n}.
-\Delta u(x)= \begin{cases}O\left( \left(\frac{1}{|x|}\right)^{(n-2)\lambda+\alpha}\right), \frac{b}{1-a}<n-2 O\left( \left(\frac{1}{|x|}\right)^{\frac{b\lambda}{1-a}+\alpha+\varepsilon}\right), \frac{b}{1-a}\ge n-2 \end{cases}
\mathbf{n}\left(E_3\right)=-\left(\frac{1}{W^{(q)}(b)}\frac{W^{(q)}(b-a)}{W^{(q)}(-a)}-\frac{1}{W^{(q+r)}(b)}\frac{W^{(q,r)}_a(b)}{W^{(q)}(-a)}\right).
\|u\|_Y :=\|u\|_{L^2}+\|\langle x\rangle^{-\frac{1+\mu}{2}}E_{1/2}u\|_{L^2}, \|f\|_{Y'}:=\inf_{f=f_1+f_2}\left(\|f_1\|_{L^2}+\|\langle x\rangle^{\frac{1+\mu}{2}}E_{-1/2}f_2\|_{L^2}\right).
u^{-\frac{1+s}{1-s}}f(u)= f(u)\cdot\frac{1-s}{1+s}\int_u^{+\infty}t^{-2/(1-s)}dt\leq \int_u^{+\infty}f(t)t^{-2/(1-s)}dt\xrightarrow[u\uparrow+\infty]{}0.
m \geq -(1-\tau)\cdot \frac{\log \delta_0}{\log(a\varepsilon)}+ \frac{\log r}{\log(a\varepsilon)}> -(1-\tau)\cdot \frac{\log r}{\log(a\varepsilon)}+ \frac{\log r}{\log(a\varepsilon)}() = \tau\cdot \frac{\log r}{\log(a\varepsilon)}.
\delta{\cal E}_{2}^{1 , 0}= \frac{8 K}{a_{0}^{6}}\left\{\left[ \left( \frac{a_{0}}{\Lambda}\right)^{2}- 2 \left( \frac{a_{0}}{\Lambda}\right) \right] e^{- 2 \Lambda / a_{0}}+ 4 E_{1}\left( \frac{2 \Lambda}{a_{0}}\right) \right\}{.}
{II_{0}\leq \prod_{i=1}^{2}\left(\frac{n_{i}(p-1)}{n_{i}(p-1)-\gamma_{i}}\right)^{\frac{1}{p^{\prime}}+\frac{1}{q}}n_{i}^{\frac{1}{p}-\frac{1}{q}-1}A_{i}\left(\int_{0}^{\infty}\int_{0}^{\infty}g^{p}(s_{1},s_{2})s_{1}^{n_{1}-1}s_{2}^{n_{2}-1}ds_{1}ds_{2}\right)^{\frac{1}{p}}.}
\ell(\mathbf{\theta})= -\log \mathrm{B}(\alpha,\beta)-\log\sigma+(\alpha-1)\log\Phi_s\left(\frac{x-\mu}{\sigma}\right) &+(\beta-1)\log\left[1-\Phi_s\left(\frac{x-\mu}{\sigma}\right)\right]+\log\phi_s\left(\frac{x-\mu}{\sigma}\right).
\int_{\mathbb{R}^n}\mathrm{div}\Phi(x) f(x)\, dx =\int_{\mathbb{R}^n}\mathrm{div}\Phi(x) (f(x) - P_tf(x))\, dx + \int_{\mathbb{R}^n}\mathrm{div}\Phi(x) P_tf(x)\, dx.
{\rm{Var}}[f_{n, j}(t)] = \sum_{i = 2}^{j + 1}\tbinom{j + 1}{i}\tbinom{n}{j + 1}\tbinom{n - j - 1}{j + 1 - i}\left[p^{2\binom{j + 1}{2}- \binom{i}{2}}- p^{2\binom{j + 1}{2}}\right].
S = \,{\frac{8 \pi^{2}}{g^{2}}}+ 2 \pi^{2}\rho^{2}( | v_{+}|^{2}+ | v_{-}|^{2}) -{\frac{i}{\sqrt{2}}}\left( \begin{array}{l l l l}{{\bar{v}_{+}}}&{{}}\\{{}}&{{\bar{v}_{-}}}\\ \end{array}\right)_{f}^{\, \, \dot{\beta}}\mu_{\dot{\beta}}\, ({\cal K}_{f}+ \tilde{\cal K}_{f}) \ ,
\langle A \rangle_{0}= \lim_{R \rightarrow \infty}\frac{1}{Z_{R}[ 0 ]}\mathrm{T r}( e^{- \beta H_{R}}\delta ( Q_{R}) A ) = \lim_{R \rightarrow \infty}\frac{1}{Z_{R}[ 0 ]}\mathrm{T r}( e^{- \beta H_{R}}P_{s}A ) ,
\Gamma_{a b c}^{\hspace{0 . 2 c m}\mu}( p_{1}, k , - p_{2})_{G V{\overline{{{G}}}}}= f_{a b c}\left[ p_{1}^{\mu}\tilde{F}_{( f ) 0}+ k^{\mu}\tilde{F}_{( f ) 1}\right] + d_{a b c}\left[ p_{1}^{\mu}\tilde{F}_{( d ) 0}+ k^{\mu}\tilde{F}_{( d ) 1}\right] \, ,
\dot{\tilde{x}}(t) = \tilde{A}(\varepsilon) \tilde{x}(t) + \tilde{B}u(t) + \displaystyle \sum_{j=1}^{n_{\rm in}}u_j(t) \tilde{N}_j \tilde{x}(t) + \tilde{H}( \tilde{x}(t) \otimes \tilde{x}(t) ) \tilde{y}(t) = \tilde{c}^\top \tilde{x}(t)
\begin{aligned}P_a^{{\varepsilon}_a}(t)e^{-itH}f \sim e^{-itH_a}\Omega_af= e^{-itH_a}P^a\Omega_af\oplus e^{-itH_a}(I-P^a)\Omega_af &\sim e^{-itH}W_a\Omega_af+e^{-itH_a}(I-P^a)\Omega_af\end{aligned}
\int_M\frac{|u_i|^2_{h^{\otimes m-l}}}{(\Sigma_{i=0}^N |s_i|^2_{h^{\otimes l}})^{n+k}}e^{-u}{\rm dv}_g\leq \frac{n+k}{k}\int_M\frac{|s|^2_{h^{\otimes m}}}{(\Sigma_{i=0}^N |s_i|_{h^{\otimes l}}^2)^{n+k+1}}e^{-u}{\rm dv}_g.
P(F)=\limsup_{n\to \infty}\bigg(h_{\mu_n}(g)+\int Fd\mu_n\bigg) \le |\mu|(h_{\mu/|\mu|}(g)+\int Fd\mu/|\mu|)+(1-|\mu|)\delta_\infty &\le |\mu|P(F)+(1-|\mu|)\delta_\infty.
\tilde{F}(x_1,\cdots,x_{k-1},x_k)=\prod_{l=0}^{k-2}\prod_{i=l+1}^{k-1}(\xi^l_{i+1}-\xi_i^l)\partial_2\partial_3^{2}\cdots \partial^{k-2}_{k-1}\partial^{k-1}_k \tilde{F}(\xi_1^0,\xi_2^1,\cdots,\xi^{k-2}_{k-1},\xi^{k-1}_k),
w_n''(t) = \begin{cases}w_{n-1}''(t) + q_{-}'(t) t < x_n - R_n, 0 x_n - R_n < t < x_n - \tau_n, \tilde{w}_n''(t) x_n - \tau_n < t < x_n, \x_n < t <x_n + \tau_n, 0 x_n + \tau_n < t < x_n + R_n, w_{n-1}''(t) + q_{+}'(t) x_n + R_n < t.\end{cases}
K_B=\frac{\lambda_2}{\lambda_1}=\left(\frac{ik-2\cos\rho}{ik+2\cos\rho}\right)^2\ \frac{ik-2}{ik+2}\ \frac{ik-2c_+}{ik+2c_+}\ \frac{ik+2c_-}{ik-2c_-},
f_{{\bf{k}},{\bf{q}}}({\bf{x}}) = e^{i{\bf{q . x}}}( \frac{1}{2 \mathrm{}N \mathrm{}\epsilon_{{\bf{q}}}}) \Lambda_{{\bf{k}}}( -{\bf{q}}) \omega_{{\bf{k}}}({\bf{q}}) + i \mathrm{}U_{-{\bf{q}}}({\bf{x}}) \Lambda_{{\bf{k}}}( -{\bf{q}})
{\mit \Sigma}(k^{2})={\mit \Sigma}(0)={-i\lambda\over2}\int\!\!{d^{4}l\over(2\pi)^{4}}{i\over l^{2}-\mu^{2}}\left(-\Lambda^{2}\over l^{2}-\Lambda^{2}\right)^{2}={-i\lambda\over2(4\pi)^{2}}\mu^{2}\left[{\Lambda^{2}\over\mu^{2}}-{\rm ln}\left({\Lambda^{2}\over\mu^{2}}\right)+1\right]\,.
k^2 m^2 n^2 - 3k m^2 n = k m^2 n (kn-3) \geq 0, k^2 m^2 n^2 - 3k m n^2 = k m n^2 (mk-3) \geq 0, 2 k^2 m^2 n^2 -3 k^2 m n - 3kmn - k^2 = k(k(mn(2mn-3)-1)-3mn).
\chi_{_K}= \left( \begin{array}{c c c}{{- ( 1 - Z ) -{\frac{4 M_{K}^{2}a^{2}\cos^{2}\theta}{\rho^{2}( 1 - Z )}}}}&{{0}}&{{-{\frac{2 M_{K}a \cos \theta}{\rho ( 1 - Z )}}}}\\{{}}&{{}}&{{}}\\{{0}}&{{1}}&{{0}}\\{{}}&{{}}&{{}}\\{{-{\frac{2 M_{K}a \cos \theta}{\rho ( 1 - Z )}}}}&{{0}}&{{-{\frac{1}{1 - Z}}}}\\ \end{array}\right) .
{\cal L}= - \frac{1}{4}F^{\mu \nu}F_{\mu \nu}- \frac{\kappa}{2}\epsilon^{\mu \nu \rho}A_{\mu}\partial_{\nu}A_{\rho}+ \sum_{a}\bar{\psi}_a [ \gamma_a^{\mu}(i\partial_{\mu}+q_a A_{\mu})- m_a] \psi_a \,\,,
\Sigma_{11}(t)-\hat{\Sigma}_{1}(t) =\frac{1}{2}\Bigg[\frac{1}{a-2k}\Big(e^{2(a-2k)t}-e^{2\lambda_+ t}\Big)+\frac{1}{a}\Big(e^{2at}-e^{2\lambda_+ t}\Big)\Bigg] &=\frac{1}{2}\frac{1}{a-2k}\Big(e^{2(a-2k)t}-e^{2a t}\Big) &=\frac{1}{2}\frac{1}{2k-a}e^{2at}\Big(1-e^{-4kt}\Big) &\leq \frac{2k}{2k-a}e^{2at}t & \leq \frac{k}{a^2 e},
[f_j,g_k]\bigg|_{-1}^1 =\langle \ell^n[f_j] , g_k\rangle - \langle f_j , \ell^n[g_k]\rangle=[j^n(j+1)^n-k^n(k+1)^n]\langle f_j , g_k\rangle & =-[k^n(k+1)^n-j^n(j+1)^n]\langle f_j , g_k\rangle = \langle f_j , \ell^n[g_k]\rangle -\langle \ell^n[f_j] , g_k\rangle =-[g_k,f_j]\bigg|_{-1}^1.
(\sum_{i=1}^{n}w_{i}x_{i}^{-p})^{-1/p}=(\frac{1}{\sum_{i=1}^{n}w_{i}\frac{1}{x_{i}^{p}}})^{1/p}\le(\frac{1}{\sum_{i=1}^{n}w_{i}\frac{1}{x_{i}^{q}}})^{1/q}=(\sum_{i=1}^{n}w_{i}x_{i}^{-q})^{-1/q}
W_x(0) =\prod_{D\in\mathcal{S}_{x}}(1- Q_{D,x,S}^t)\prod_{D\in\mathcal{U}_{x}}(1- Q_{D,x,U}^t)-\prod_{D\in\mathcal{S}_{x}}(1- Q_{D,x,*}^t)\prod_{D\in\mathcal{U}_{x}}(1- Q_{D,x,*}^t).
[y_{(1\,l)},y_l]_c =(1-\chi_l(g_{(1\,l)})\chi_{(1\,l)}(g_l))y_{(1\,l)}y_l-\chi_l(g_{(1\,l)})[y_l,y_{(1\,l)}]_c &=(1-\xi)y_{(1\,l)}y_l-\chi_l(g_{(1\,l)})(1-\xi)y_ly_{(1\,l)}&=(1-\xi)[y_{(1\,l)},y_l]_c.
\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^{2}}=\frac{1}{1^{2}}-\frac{1}{2^{2}}+\frac{1}{3^{2}}-\frac{1}{4^{2}}+\cdot\cdot\cdot=\frac{\pi^{2}}{12}
\mathcal{R}_{1}(n) =4\sum_{d|n}\left(\frac{-4}{d}\right), \mathcal{R}_{2}(n) =8\sum_{d|n}d-32\sum_{d|\frac{n}{4}}d, \mathcal{R}_{3}(n) =-4\sum_{d|n}\left(\frac{-4}{d}\right)d^{2}+16\sum_{d|n}\left(\frac{-4}{n/d}\right)d^{2}, \mathcal{R}_{4}(n) =16\sum_{d|n}d^{3}-32\sum_{d|\frac{n}{2}}d^{3}+256\sum_{d|\frac{n}{4}}d^{3}
H_{X^M}= \left[g,\pi_{\mathfrak m}\left(g^{-1}dg\right)\right]d\pi dR_g X=\left[g, \pi_{\mathfrak m}\left(g^{-1}dg\right)\left(g^{-1}X g\right)_{\mathfrak m}\right] = \left[g, \pi_{\mathfrak m}\left(g^{-1}X g\right)\right]=X_M.
\int_{1}^{n}\frac{\psi\left( x+1\right)}{x^{p}}dx =\frac{\gamma}{p-1}\left( \frac{1}{n^{p+1}}-1\right) +\sum_{k=1}^{\infty}\sum_{j=2}^{p-1}\frac{\left( -1\right)^{p-1-j}}{k^{p+1-j}}\frac{1}{j-1}\left(1-\frac{1}{n^{j-1}}\right) & +\sum_{k=1}^{\infty}\frac{\left( -1\right)^{p}}{k^{p}}\left( \ln\frac{n}{k+n}-\ln\frac{1}{k+1}\right) .
\Pi^1_{\mu_1\mu_2}(k,u)=\frac{1}{2\pi i}\int\frac{{\rm d}^3 Q}{\left(2\pi\right)^3}\int_{\cal{C}}{\rm d Q_0}N\left( Q_0\right)t^{\alpha\beta\rho\sigma}_{\mu_1\mu_2}\left(Q_0,{\bf Q}, Q_0+k_0,{\bf Q}+{\bf k}\right)D_{\alpha\beta}(Q) D_{\rho\sigma}(Q+k),
LR({\mathbb T}) :=L(g_1)({\mathbb T}|_{X^{g_1,g_2}}) + L(g_2)({\mathbb T}|_{X^{g_1,g_2}}) + L((g_1g_2)^{-1})({\mathbb T}|_{X^{g_1,g_2}}) -{\mathbb T}|_{X^{g_1,g_2}}+ ({\mathbb T}|_{X^{g_1,g_2}})^{g_1,g_2}.
\begin{gathered}N_k^{n}=(b^{2^{k}}-1)^{n}=\sum_{i=1}^{\frac{n+1}{2}}\Big ( C_{n}^{n+2-2i}b^{(n+2-2i)\cdot 2^{k}}-C_{n}^{n+1-2i}b^{(n+1-2i)\cdot 2^{k}}\Big ).\end{gathered}
N_{p_{D}n}= | g ( - |^{+}) |^{2}= e^{- \pi \lambda}, \; \; \mathrm{i f}\quad \sqrt{\rho}\, T \gg 1 , \; \; \mathrm{a n d}\; \sqrt{\rho}T \gg \lambda , \; \mathrm{a n d}\; \rho^{2}T \gg | q E p_{D}| \; ,
L_7^{-1}=(1-T)( 7^4T^8 - 588T^6 + 134T^4 - 12T^2 + 1), L_{227}^{-1}=(1+T) (\ldots + 200T^2 + 13T + 1), L_{1277}^{-1}=(1+T) (\ldots-35T+1), L_{1609}^{-1}=(1+T) (\ldots-26T+1).
W^{( 2 )}( x ; B^{\prime}, \psi_{0}, \psi_{1}) = \frac{1}{1 2}( B^{\prime}x_{0}^{1 2}+ x_{3}^{1 2}) + \frac{1}{3}x_{4}^{3}+ \frac{1}{2}x_{5}^{2}- \psi_{0}\, x_{0}x_{3}x_{4}x_{5}- \frac{1}{6}\psi_{1}\, x_{0}^{6}x_{3}^{6}\ ,
U_{\mathrm{{s y m . t r a c e l e s s}}}^{\mu \nu \lambda \rho \sigma}= \mathrm{s y m m e t r i z a t i o n o f}U^{\mu \nu \lambda \rho \sigma}- \frac{1}{4 0}( g^{\mu \nu}U_{\alpha}^{\alpha \lambda \rho \sigma}+ \mathrm{9 t e r m s}) .
v_{m}( z ) = \frac{\sqrt{m / 2 k}\left[ J_{1}( m / k ) Y_{2}( m e^{k | z |}/ k ) - Y_{1}( m / k ) J_{2}( m e^{k | z |}/ k ) \right]}{\sqrt{J_{1}( m / k )^{2}+ Y_{1}( m / k )^{2}}}.
\sum_{d \in \mathbb{Z}}\mathrm{Tr}_{d,D}^{\mathrm{add}}(F_{0,m,r})q^d := \mathrm{LC}_{s=1}^r[(\ref{finpart})]^{\mathrm{hol}}= \sum_{n|m}\biggl(\frac{D}{m/n}\biggr) [(\ref{finhol}) + (\ref{finhol2})].
N_m(v) \leq \ \underbrace{C' \sum_{\substack{1 \leq a_1 \leq \dots \leq a_k \leq m-1 a_1+ \dots+a_k = m+1}}\int_{S^1}{| v^{(a_1)}\dots v^{(a_k)}v^{(m)}|}}_{N_{m,1}}+\underbrace{C' \int_{S^1}{\left|(v^{(m)})^2 v' \right|}}_{N_{m,2}}.
0\leq \left\langle x-\sum_{j\in \mathbb{J}}A_j^*A_jx ,x-\sum_{k\in \mathbb{J}}A_k^*A_kx \right\rangle =\langle x, x \rangle-2\sum_{j\in \mathbb{J}}\langle A_jx, A_jx \rangle +\sum_{j\in \mathbb{J}}\langle A_jx, A_jx \rangle =\langle x, x \rangle-\sum_{j\in \mathbb{J}}\langle A_jx, A_jx \rangle,
{_{s+1}}F_s\left(\left\{1\right\}^{s},1-n;\left\{2\right\}^{s};1\right)=\sum\limits_{k=1}^n(-1)^{k-1}\binom{n}{k}H_k(s), &{_{s+1}}F_s\left(\left\{1\right\}^{s},1-n;\left\{2\right\}^{s};-1\right)=\sum\limits_{k=1}^n(-1)^{k-1}\binom{n}{k}H_k(\overline{s}).
\frac{(a_{4}^{*})^{1/3}}{\sqrt{p}}\sum_{k=1}^{\infty}k^{3/2}(131958.36 (4k^{3}+ 6k^{2}+ 4k + 1) + 263919.6 (k+1)^{4}) M(25.09 (k+1)^{2}p^{35/6}) e^{-4 \pi k}.
\int \left| Q(\nabla)G_{h/2}\ast u\right|^2 dx = \int \left| \int Q(\nabla)G_{h/2}(z) \left(u(x-z)-u(x)\right) dz\right|^2dx \lesssim \int \left| Q(\nabla)G_{h/2}(z)\right| dz \int \left|Q(\nabla)G_{h/2}(z)\right| \left| u(x-z)-u(x)\right|^2\, dz\,dx.
\sqrt{\gamma_1}\, \sqrt{\gamma_2}\C_{\rm tot}(X_1 , X_2) = 32 \, G_N\, T^2 \, (\partial_+ \, X_1^{\mu}\ \partial_+ \, X_{2 \mu})(\partial_- \, X_1^{\nu}\ \partial_- \, X_{2 \nu}) \, .
[(g_\alpha\eta_{N,w})^{(k)}(z)]_{k=0}^{k_\lambda-1}=g_\alpha(z)\cdot\Psi_{k_\lambda}(z)\cdot[\kappa_{k_\lambda,k}(z)]_{k=0}^{k_\lambda-1}= \omega_a(z)\cdot[\psi_{\omega,k}(z)]_{k=0}^{k_\lambda-1}= [\omega_a^{(k)}(z)]_{k=0}^{k_\lambda-1}\ \ \z=\beta_w(\lambda), \ \lambda\in\Lambda.
\lim_{\eta \rightarrow 0}\lim_{\Lambda}\omega_{\beta,{\mu}_{\Lambda}(\beta,{\rho}, \eta),\Lambda,\eta}^{0}({b_{q}}/\sqrt{V}) =\lim_{\eta \rightarrow 0}\frac{\eta}{\mu(\beta,{\rho}, \eta)}= e^{i \,{\rm{arg}}(\eta)}\, \sqrt{{\rho}- \rho_{c}(\beta)}\ ,
\frac{K(t,s)}{K(u,s)}=\frac{\frac{1}{1-s+t}+\frac{1}{(\lambda_{0}(-t)^{1-\alpha}-s+t)(1-s+t)^{3}}}{\frac{1}{1-s+u}+\frac{1}{(\lambda_{0}(-u)^{1-\alpha}-s+u)(1-s+u)^{3}}}&\leq \frac{1-s+u}{1-s+t}+ \frac{(\lambda_{0}(-u)^{1-\alpha}-s+u)(1-s+u)^{3}}{(\lambda_{0}(-t)^{1-\alpha}-s+t)(1-s+t)^{3}}&\leq 2, s \leq u \leq t
\hat{S}_{1T}\left( u\right) =S_{1T}\left( u\right) +T^{1/2}\left( \hat{\theta}_{T}-\theta_{0}\right)^{\prime}\frac{1}{T}\sum_{t=1}^{T}\nabla\left(F_{t,\theta_0}\left(\cdot \mid \Omega_t\right),u\right) +o_{p}\left(1\right) ,
\begin{array}{rl|rl}L_2L_{-2}\Psi =\ (4h+c/2)\Psi L_2L_{-1}{}^2\Psi =\ 6h\Psi L_{1}{}^2L_{-2}\Psi =\ 6h\Psi L_{1}{}^2L_{-1}{}^2\Psi =\ 4h(2h+1)\Psi\,,\end{array}
e^{-s(k^* - i) t}e^{-s(j - k^*)(t - \tau_j)}= e^{-s(j-i)(t - \tau_j)}e^{-s(k^* - i)\tau_j}&\leq e^{-s(j - i)(t - \tau_j)}k_N^{-\theta(k^* - i)}&= \bigg( \frac{\mu}{s}\bigg)^{j - i}e^{-s(j - i)(t - \gamma_j)}k_N^{-\theta(k^* - i)}.
A^{(\ref{sdir})}(p,\delta)=1 - \frac{g_0}{9}+ \frac{g_0}{3p^2}\delta^2 -\frac{g_0}{3p^3}\delta^3 \arctan\left(\frac{p}{\delta}\right)+ \frac{g_0}{6}ln\left(\frac{p^2 + \delta^2}{\alpha^2 + \delta^2}\right).
\sum_{(A,WB)=1}\log{A}\frac{\omega(A)}{\phi_\omega(A)^2}\sum_{\substack{\mathbf{r}\in\mathcal{D}'_k r_m=1 A|r}}\frac{Y_{\mathbf{r}}^2\phi(r/A)}{\phi_\omega(r/A)^2}\ll \Bigl(\sum_{(A,WB)=1}\log{A}\frac{\omega(A)^2}{\phi_\omega(A)^2}\Bigr)\sum_{\substack{\mathbf{r}'\in\mathcal{D}_k r'_m=1}}\frac{Y_{\mathbf{r}'}^2\phi(r')}{\phi_\omega(r')^2}.