harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left| \overline{Q_l}(s,y) \right| = \left| Q\left(I^{-1}\frac{y_1 - l_1 s}{\sqrt{1-l_1^2}}, I^{-1}y_2, I^{-1}y_3\right) \right| &\le K(Q) I \left( \left( \frac{y_1 - l_1 s}{\sqrt{1-l_1^2}}\right)^2 + y_2^2 + y_3^2 \right)^{-\frac12}&\lesssim K(Q) I |y|^{-1}.
M_{1}(\mathcal{NC}(D_{2m})) =(2m-1)(2m-2)^{2}-4(2m-2)\dfrac{(m-1)(m-2)}{2}+(m-1)(m-2)^{2}= (m-1)[(8m-4)(m-1)-4(m-1)(m-2)+(m-2^{2})] = m(m-1)(5m-4)
deg \big(|0\rangle \big) =0; deg \chi_{-j}|0\rangle =j; deg \Big(\left(\chi_{-j_k}\right)^{m_k}\dots \left(\chi_{-j_2}\right)^{m_2}\left(\chi_{-j_1}\right)^{m_1}|0\rangle \Big) =m_k\cdot j_k +\dots m_2\cdot j_2 +m_1\cdot j_1,
W_{2p}=\frac{\Gamma(p+\frac{1}{2})\Gamma(\frac{1}{2})}{2\Gamma(p+1)}=\frac{(2p-1)!!\pi}{2^{p+1}p!}=\frac{(2p)!}{2^{2p}(p!)^{2}}\cdot\frac{\pi}{2}.
g_t (x) = 2^{t}\cdot \left( \varrho (x) + \sum_{k=1}^{2^{t-1}- 1}2 \cdot \varrho\left( x - \frac{2k}{2^{t}}\right) - \sum_{\ell=1}^{2^{t-1}}2 \cdot \varrho \left( x - \frac{2\ell - 1}{2^t}\right) \right) x \in [0,1] \, .
\sigma^{(1)}_{{\rm tot}\,i}= \frac{(2\pi)^4\,\hbar^2}{v_i}\frac{1}{\pi}{\rm Im}R^{(1)}_{ii}\exp \left[-\frac{1}{\hbar}\int^{+\infty}_{-\infty}dt\, \Gamma^{(1)}(t) \right]\,.
P(L_{n+m}\ge y+z-c(nm)^{\frac{1}{d+1}}) \ge P(L_{n+dq+m}\ge y+z-(c-d)(nm)^{\frac{1}{d+1}}) \ge P(L_{n+p+m}\ge y+z-(c-d-2d^2)(nm)^{\frac{1}{d+1}}) \ge P(L_{n}\ge y,\hat{L}_m\ge z, S_{n+p}-S_n=\lambda,N_{n,m}(\lambda)\le \frac{1}{4d}(c-d-2d^2)(nm)^{\frac{1}{d+1}}).
\widetilde \Omega = \Omega_{\mbox{\tiny{$\widetilde{Q}$}}}- B_{\mbox{\tiny{$\!\langle\! J\mathcal{K}\!\rangle$}}}= -d(p_\theta d\theta+p_\psi d\psi + p_\phi d\phi) + r\tfrac{F(\phi)}{\sin^2\phi}(p_\theta-p_\psi) d\theta\wedge d\phi,
S=\int Tr\partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}+f_{jk}^{i}\partial^{\nu}A_{i}^{\mu}A_{\mu}^{j}A_{\nu}^{k}+f_{jr}^{i}f_{kl}^{r}A_{i}A_{j}A^{k}A^{l}+Tr\partial_{\mu}\overline{\eta}\partial^{\mu}\eta+\overline{\eta}A_{j}\eta
\alpha_{- n}^{-}= - \frac{1}{4}\sum_{l = - \infty}^{+ \infty}\left[ \alpha_{n - l}^{i}, \alpha_{l}^{i}\right]_{+}- \frac{1}{4}\sum_{r = - \infty}^{+ \infty}\left( r - \frac{n}{2}\right) \left[ b_{n - r}^{i}, b_{r}^{i}\right] - \frac{a}{2}
\frac{d}{dt}\int_\mathbb{R}\sum_{i=1}^n\big(a^2 w_{ix}^2 + w_{it}^2\big) dx+2 \int_\mathbb{R}\sum_{i=1}^n \big[w_{it}^2+\lambda_i(u^a) w_{it}w_{ix}\big] dx= \rm{I}_3 + \rm{I}_4,
[I_a \mathbf{v}]_{j}\leq \left( \mathop{\sum_{l=1}^{r+d}}_{l \in I_{k+1}}v_l \right) - v_{i}= \sum_{l=1}^{n_{k+1}}\left( \frac{V_k}{n_{k+1}}- \delta \right) -V_k = \left( V_k - n_{k+1}\delta \right) - V_k = -n_{k+1}\delta < 0.
\log |A| \leq \log d & \leq \sum_{p \leq n + 1}M(n,p) \log p + \sum_{p > n+1}M(n,p) \log p & \leq n \sum_{p \leq n + 1}\left( \frac{1}{p}+ O\left(\frac{1}{p^2}\right) \right) \log p + \sum_{p > n+1}0 \log p &= n \log n + O(n).
\left( \begin{matrix}{{1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{1}}&{{- \sqrt{2}\eta}}&{{\sqrt{2}\eta}}\\{{0}}&{{\sqrt{2}\eta}}&{{1 - \eta^{2}}}&{{\eta^{2}}}\\{{0}}&{{\sqrt{2}\eta}}&{{- \eta^{2}}}&{{1 + \eta^{2}}}\\ \end{matrix}\right) .
|S(x)|^2_{\widetilde{h}_p}= |s(0)|^2 \leq \frac{\int_{\Delta^n(0,r_p)}|s|^2 e^{-2p \psi^\prime_x}\,dm}{\int_{\Delta^n(0,r_p)}e^{-2p \psi^\prime_x}\,dm}\,\cdot
\frac{P_{01r}(N)}{(-4)^r r!}= \left[ z^{r}\right]e^z \left(\frac{z}{e^z - 1}\right)^{r+1}\exp\left( \sum_{j=1}^r \frac{(-1)^{j+1}B_j z^j}{j \cdot j!}\Bigl(s_j(N)-N \Bigr)\right).
|C_k| U(\bar p) = |C_k|U(\bar p_j,\bar p_{-j}) =|C_k| U(\left[ |C_k|^{-1}\sum_{i\in C_k}p_i\right]_j,\bar p_{-j}) & = \sum_{i\in C_k}U([p_i]_j,\bar p_{-j}) = \sum_{i\in C_k}U(p_i,\bar p_{-i}),
\left| \bigcap_{i\in I}A_i \right| = \left| \bigcap_{j\in I}\overline{B_j}\right|= \sum_{J\subseteq I}(-1)^{|J|}\left| \bigcap_{j\in J}B_j \right| = \sum_{J\subseteq I}(-1)^{|J|}\frac{n}{\prod_{i\in I}i \prod_{j\in J}j^{n_j-1}}.
{}\bar{\pi}_{h,{T_n}}(x) = \frac{1}{T_n \prod_{l = 1}^d h_l}\int_0^{T_n}\prod_{m = 1,2}K(\frac{x_m - X_u^m}{h_m}) \prod_{l = 3}^d K(\frac{x_l - X_{\varphi_{n, l}(u)}^l}{h_l}) du = : \frac{1}{T_n}\int_0^{T_n}\prod_{m = 1,2}K_{h_m}(x_m - X_u^m) \prod_{l = 3}^d K_{h_l}(x_l - X_{\varphi_{n, l}(u)}^l) du.
T= \sum_{n=0}^\infty\frac{1}{n!}\left(\sum_{k=0}^n \binom{n}{k}\frac{1}{k!}(-1)^{n-k}\right)x^nD^n=1-\frac{1}{2}x^2D^2+\frac{2}{3}x^3D^3-\frac{5}{8}x^4D^4+\cdots.
\int_{a}^{x}f ( \hat{x}) \, ( d \hat{x})^{n}\, \equiv \, \underbrace{\int_{\! a}^{x}\! \int_{\! a}^{x_{n}}\! \! \int_{\! a}^{x_{n - 1}}\! \! \! \! \! \! \! \! \cdots \! \int_{\! a}^{x_{3}}\! \! \int_{\! a}^{x_{2}}\!}_{n \mathrm{\footnotesize - t i m e s}}f ( x_{1}) \; d x_{1}\, d x_{2}\cdots d x_{n - 1}\, d x_{n}
T^{c}\bigl ( A_{1}( x_{1}) . . . A_{n}( x_{n}) \bigr ) = ( A_{\pi 1}( x_{\pi 1}) \times_{\hbar}. . . \times_{\hbar}A_{\pi n}( x_{\pi n}) )^{c}={\cal O}( \hbar^{n - 1}) .
\frac{\big(x\cdot t\cdot t^2\cdots t^{N-1}\big)^m \cdot P_{\lambda}\big(x, t, t^2, \dots, t^{N-1}; q, t\big)}{\big(1\cdot t\cdot t^2\cdots t^{N-1}\big)^m \cdot P_{\lambda}\big(1, t, t^2, \dots, t^{N-1}; q, t\big)}= \frac{P_{\widetilde{\lambda}}\big(x, t, t^2, \dots, t^{N-1}; q, t\big)}{P_{\widetilde{\lambda}}\big(1, t, t^2, \dots, t^{N-1}; q, t\big)}.
\|uC_\psi\|_e\simeq \begin{cases}\bigg(\limsup_{|w| \to\infty}B_{(\psi,m)}(|u|^q)(w) \bigg)^{\frac{1}{q}}, q<\infty \limsup_{|\psi(w)| \to\infty}B^\infty_{(\psi,m)}(|u|)(w), q= \infty.\end{cases}
\delta{\widetilde A}_{0 1}= - \frac{\mathrm{e}^{\varphi_{0}}\, ( m - \chi_{0}\, n )}{{\widetilde \Delta}_{m , n}^{1 / 2}}\, \frac{{\widetilde Q}_{1}}{r^{6}}, \delta{\widetilde C}_{0 1}= - \frac{\mathrm{e}^{\varphi_{0}}\, ( | \lambda_{0}|^{2}\, n - \chi_{0}\, m )}{{\widetilde \Delta}_{m , n}^{1 / 2}}\, \frac{{\widetilde Q}_{1}}{r^{6}}
M(1,x)=\theta_{3}^{-2}(exp(-\pi\frac{M(1,x)}{M(1,\sqrt{1-x^{2}})}))=(\sum_{n\in\mathbb{Z}}exp(-n^{2}\pi\frac{M(1,x)}{M(1,\sqrt{1-x^{2}})}))^{-2},
\mathbb{P}_{n_{l}}\left[ -M \le \log\left(Y_{n_{l}}\right) \le M \right] \ge 1- \frac{\delta}{100}&\mathbb{P}_{n_{l}}\left[ -M \le \left\{\sum_{i=1}^{k}\frac{2\mu_{i}W_{n_{l},i}- \mu_{i}^2}{2 \sigma_{i}^2}\right\}\le M \right] \ge 1 -\frac{\delta}{100}
A_m(x) =B_{m-1}(x)+x^2A_{m-1}(x) =xA_{m-1}(x)+B_{m-2}(x)+x^2A_{m-1}(x) =xA_{m-1}(x)+A_{m-1}(x)-x^2A_{m-2}(x)+x^2A_{m-1}(x) =(1+x+x^2)A_{m-1}(x)-x^2A_{m-2}(x).
V ={x^{p - 2}}\,{y^{p - 2}}-{x^{p - 4}}\,{y^{p - 4}}\, \left({x^{3}}+{y^{3}}\right) +{x^{p - 6}}\,{y^{p - 6}}\, \left({x^{6}}+{y^{6}}\right) + \cdots + x \, y \, \left({x^{3 \,{\frac{p - 3}{2}}}}+{y^{3 \,{\frac{p - 3}{2}}}}\right)
{\frac{d}{d t}}\left( \begin{matrix}{p}\\{q}\\ \end{matrix}\right) ={\frac{1}{\hbar}}\left( \begin{matrix}{0}&{-{\cal H}}\\{{\cal H}}&{0}\\ \end{matrix}\right) \left( \begin{matrix}{{\frac{\delta H_{0}}{\delta p}}}\\{{\frac{\delta H_{0}}{\delta q}}}\\ \end{matrix}\right)
E=(\frac{2e^{2}}{L_{B}})\int_{0}^{\infty}\frac{kdk}{k^{2}+k_{B}^{2}r_{B}^{2}}M(l+1,1,-\frac{k^{2}}{4})M(l^{\prime}+1,1,-\frac{k^{2}}{4})J_{0}(k\frac{r_{12}}{r_{B}})
s_j:=[u^j]S(z,u) ={\frac{{z}^{4}+{z}^{4}g_0+{z}^{2}g_0-{z}^{2}+1}{zr_1(1-u/r_1)}}&={\frac{{z}^{4}+{z}^{4}g_0+{z}^{2}g_0-{z}^{2}+1}{zr_1^{j+1}}}.
[\begin{matrix}t\\u\\v\end{matrix}]=\frac{1}{-l_{ab}\cdot(p_{01}\times p_{02})}[\begin{matrix}{(p_{01}\times p_{02})}^{T}\\{(p_{02}\times-l_{ab})}^{T}\\{(-l_{ab}\times p_{01})}^{T}\end{matrix}][l_{a}-p_{0}]
D(P\|Q) = \int_{\mathcal{A}}Q \, \exp\bigl(i_{P \| Q}\bigr) \, i_{P \| Q}begin{align*}0.1cm] = \int_{\mathcal{B}}Q \, \exp\bigl(i_{P \| Q}\bigr) \, i_{P \| Q}+\int_{\mathcal{A}\setminus \mathcal{B}}Q \, \exp\bigl(i_{P \| Q}\bigr) \, i_{P \| Q}.
\phi_{ij}(x)=\frac{e^x-1}{\lambda e^x-\mu},\phi_{k\ell}(x)=-\frac{e^x-1}{(\mu+\alpha)e^x-(\lambda+\alpha)},\phi_{i\ell}(x)=\frac{(\lambda-\mu-\alpha) e^x}{1-\lambda(\mu+\alpha)e^x}.
\begin{array}{l}\{|H_{13}(H_{22}-\lambda_{j})-H_{23}H_{12}|^{2}+ |H_{23}(H_{11}-\lambda_{j})-H_{13}H_{21}|^{2}+ \left[|H_{12}|^{2}-(H_{11}-\lambda_{j})(H_{22}-\lambda_{j})\right]^{2}\}^{-1}\times\end{array}
\begin{pmatrix}\phi_{11}0 \phi_{21}\phi_{22}\end{pmatrix}=\begin{pmatrix}\pi^{-1}_{11}\phi_{11}\pi_{11}0 -\pi^{-1}_{22}\pi_{21}\pi^{-1}_{11}\phi_{11}\pi_{11}+\pi^{-1}_{22}\phi_{22}\pi_{21}\pi^{-1}_{22}\phi_{22}\pi_{22}\end{pmatrix}
[ X^{\#_{I^\varpi_j},[\mathbf{u}^{(1)}]^\varpi_j}_t + \lambda X^{\#_{I^\varpi_j},[\mathbf{u}^{(2)}]^\varpi_j}_t ]_{\P,\mathcal{B}(H)}= [ X^{\#_{I^\varpi_j},[\mathbf{u}^{(3)}]^\varpi_j}_t ]_{\P,\mathcal{B}(H)}.
c_{\mu}^{(s)}(\rho)=\frac{i\sqrt{\sinh\pi\mu}}{2\pi}\,\phi_M(0,\rho)\left\{\Gamma(-i\mu)\left(\frac{m\rho}2\right)^{i\mu}-\Gamma(i\mu)\left(\frac{m\rho}2\right)^{-i\mu}\right\},
f_{0}( a , t ) = 2 \sum_{\alpha \in \mathbf{C}}\frac{\sinh ( b a_{\alpha}t ) \, \sinh ( ( b a_{\alpha}- 2 b Q_{\alpha}) t )}{\sinh 2 t \sinh ( \alpha^{2}b^{2}t ) \, \sinh ( ( 1 + b^{2}) 2 h t )},
{\frac{m}{( 2 \pi )^{3}}}\int \left[ ( g_{1}^{( 1 )})^{2}+ ( g_{2}^{( 1 )})^{2}+ ( g_{3}^{( 1 )})^{2}+ ( g_{4}^{( 1 )})^{2}\right]{\frac{d^{3}k}{\varepsilon_{k}}}= 1 .
\langle \phi ( z , \bar{z}) \phi ( z^{\prime}, \bar{z}^{\prime}) \rangle ={\frac{1}{4 \pi}}\int_{0}^{\infty}{\frac{d s}{s}}e^{-{\frac{d^{2}}{4 s}}- m^{2}s}={\frac{1}{2 \pi}}K_{0}( d^{2}m^{2}) ,
\delta^{\Psi\Psi}_{ppq(q-1)j}= \frac{\dim \Psi_{pqj}}{\dim \Psi_{p(q-1)j}}\biggl( 1-\frac{\dim \Psi_{(p+1)qj}}{\dim \Psi_{pqj}}\delta^{\Psi\Psi}_{p(p+1)qqj}\biggr).
\sum_{\substack{q\leq\Omega (q,Mp_0)=1}}\;\max_{(a,qM)=1}\;\Big\lvert\frac{1}{\varphi(qM)}\sum_{\substack{\chi\pmod{qM}\chi\neq\chi_0}}\overline{\chi}(a)\psi(X,\chi)\Big\rvert.
G_\mu(J_1(\tau),J_2(\tau),J_3(\tau),J_1(\tau')) = 0 \ , H_{\mu,2}(J_1(\tau),J_2(\tau),J_3(\tau),J_1(\tau'),J_2(\tau')) = 0 \ , H_{\mu,3}(J_1(\tau),J_2(\tau),J_3(\tau),J_1(\tau'),J_3(\tau')) = 0 \ .
g'_n(t_n) = (1-f(1)) \lambda_n \frac{1-f(1)-\delta}{1-f(1)}\left(\frac{1-f(1)-\delta}{1-f(1)}\right)^{-1/\lambda_n}= \lambda_n (1-f(1)-\delta) \left(\frac{1-f(1)-\delta}{1-f(1)}\right)^{-1/\lambda_n}.
\tilde{\mathbb{H}}:=\frac{1}{2N}\int\mathrm{d}y_1\mathrm{d}y_2 v_N(y_1-y_2)\mathcal{Q}^\ast_{y_1y_2}\mathcal{Q}_{y_1y_2}, & \mathbb{H}:=N^{-1/2}\mathcal{E}(t)-\tilde{\mathbb{H}}.
\tilde P_\pm^* \tilde P_\pm u[x] =(2\pi)^{-d}\int_{\mathbb{T}^d}\sum_{y\in\mathbb{Z}^d}e^{i(\varphi_a(x,\xi)-\varphi_a(y,\xi))}p_\pm(x,\xi)p_\pm(y,\xi)u\left[y\right]d\xi &=(2\pi)^{-d}\int_{\mathbb{T}^d}\sum_{y\in\mathbb{Z}^d}e^{i(x-y)\cdot\eta(\xi;x,y)}p_\pm(x,\xi)p_\pm(y,\xi)u\left[y\right]d\xi,
\begin{bmatrix}u v \end{bmatrix}\mapsto \mathrm{i}\begin{bmatrix}0 -1 1 0 \end{bmatrix}\begin{bmatrix}u' v' \end{bmatrix}+ \begin{bmatrix}0 0 0 \mathrm{i}\end{bmatrix}\begin{bmatrix}u v \end{bmatrix}
\sum_{q=1}^{\infty}{\frac{{{c_q}(n)}}{q}}=0, \sum_{q=1}^{\infty}{\frac{{\log q}}{q}{c_q}(n)}=-d(n),\quad\sum_{q=1}^{\infty}{\frac{{{{( - 1)}^{q - 1}}{c_{2q - 1}}(n)}}{{2q - 1}}}=\frac{1}{\pi}r(n)
nR_\Sigma \leq \sum_{j=1}^2\left[h(S_j^n|V_j) - h(N_j^n)+ h(Y_j^n|S_j^n,V_j)\right] &- h(\tilde{S}_{2}^n|V_{2}) - h(\tilde{S}_{1}^n|V_{1},\bar{d}X_3^n)+n\epsilon_n \overset{(c)}{\leq}\sum_{j=1}^2 \left[h(Y_j^n|S_j^n,V_j)- h(N_j^n)\right]+ n +n\epsilon_n
q^{\beta^{2}/2}\left(q^{-\beta+n+1};q\right)_{\infty}L_{n}^{(-\beta)}\left(-1;q\right)=\frac{1}{\sqrt{\pi\log q^{-2}}}\int_{-\infty}^{\infty}\frac{\exp\left(\frac{y^{2}}{\log q^{2}}+i\beta y\right)}{\left(q;q\right)_{n}\left(-q^{1/2+n}e^{-iy};q\right)_{\infty}}dy,
Z_n((G,G_1),\varphi,\gamma)\! \! =\frac{1}{m^n}\sum_{|\underline{h}_1|=n}\sup_{E}\sum_{x\in E}e^{S_{\underline h_1}\varphi(x)}\!\leq\!\frac{m^{n+k}}{m^{n}}\frac{1}{m^{n+k}}\sum_{{\underline g \in G_n^*}+k}\sup_{E}\sum_{x\in E}e^{S_{\underline g}\varphi(x)}\!=\! m^k Z_{n+k}((G,G_1),\varphi,\varepsilon).
\int_{I \times \partial Q}g\cdot n_{\eta_k}\, d|\sigma_k| -\int_{I \times \partial Q}g\cdot n_\eta \, d|\sigma| = \int_{I \times \partial Q}g\cdot (n_{\eta_k}-n_\eta) \, d|\sigma_k| + \int_{I \times \partial Q}g\cdot n_\eta \, d(|\sigma_k| - |\sigma|).
\langle 0 | ( X^{1}( \sigma ) - \overline{{X^{1}}})^{2}| 0 \rangle \ = \ \sum_{m = 1}^{\infty}\frac{1}{m^{2}}\langle 0 | ( \alpha_{m}\alpha_{- m}+ \tilde{\alpha}_{m}\tilde{\alpha}_{- m}) | 0 \rangle \ = 2 \sum_{m = 1}^{\infty}\frac{1}{m}.
\frac{\left[ \left( \begin{array}{c}{{\psi}}\\{{0}}\\ \end{array}\right) , \widetilde{A}\left( \begin{array}{c}{{\psi}}\\{{0}}\\ \end{array}\right) \right]_{{\cal K}_{2}}}{\left[ \left( \begin{array}{c}{{\psi}}\\{{0}}\\ \end{array}\right) , \left( \begin{array}{c}{{\psi}}\\{{0}}\\ \end{array}\right) \right]_{{\cal K}_{2}}}= \frac{\langle \psi \mid A \psi \rangle}{\langle \psi \mid \psi \rangle},
f_\infty^\kappa(w) = C_{\varsigma,m}(1-w)^{-2-\frac{w_d-1}{\varsigma}-\frac{w_d}{\kappa\varsigma}}(1+w)^{-2+\frac{w_d+1}{\varsigma}+\frac{w_d}{\kappa\varsigma}}\exp\left\{-\frac{2}{\kappa}\frac{1-w_d w}{\varsigma\left(1-w^2\right)}\right\}
\begin{aligned}T_3\ge -C s^2 \lambda^2 \iint_{Q}\xi^3 \left|A \nabla \eta \cdot \nabla \eta \right|^2 u^2 dx dt -C s^2 \lambda^2 \int_{0}^{T}\int_{\omega}\xi^3 u^2 dx dt,\end{aligned}
a_{D}={\frac{\sqrt 2}{\pi}}\int_{1}^{{\cal G}( a ) + B}{\frac{d x \sqrt{x -{\cal G}( a ) - B}}{\sqrt{x^{2}- 1}}}, \qquad a ={\frac{\sqrt 2}{\pi}}\int_{- 1}^{1}{\frac{d x \sqrt{x -{\cal G}( a ) - B}}{\sqrt{x^{2}- 1}}}.
\Gamma_{1,-1}= 0, ~ \Gamma_{1,0}= 1, \Gamma_{1,1}= \tau \Gamma_{2,-1}= 1, ~ \Gamma_{2,0}= 0, \Gamma_{2,1}= -\sigma \Gamma_{3,-1}= 0, ~ \Gamma_{3,0}= 0, \Gamma_{3,1}= \delta.
M_{0j}=\frac{1}{2}\big( d\Gamma(X_jP_0)+d\Gamma(P_0X_j)+x_0 d\Gamma(P_0)+ d\Gamma(P_0) x_0\big) &=\frac{1}{2}\big( d\Gamma(X_jP_0)+d\Gamma(P_0X_j)\big) +x_0 d\Gamma(P_0).
-8\pi y_{p}\bigg(-y_{p}+ 2\pi\coth\bigg(\frac{2\pi}{y_{p}}\bigg)\bigg)=-8\pi y_{p}\bigg(\frac{4\pi^{2}}{3 y_{p}}+O\bigg(\frac{1}{y_{p}^{3}}\bigg)\bigg)-\frac{32\pi^{3}}{3}+O\bigg(\frac{1}{y_{p}^{2}}\bigg),
\bar{G}= \underset{t \to \infty}{\lim}\frac{\int \phi_m\left(x\right) \, q_t(x) dP_0^m(x)}{\int \Pi_{j=1}^{m}\phi_1\left( x^j\right) \, q_t(x) dP_0^m(x)}= \frac{\int \phi_m\left(x\right) \, q(x) dP_0^m(x)}{\int \Pi_{j=1}^{m}\phi_1\left( x^j\right) \, q(x) dP_0^m(x)}.
\|C_j\|_{\Phi_{j}(\ell_j)}= \ell_j^{-2}\sup_{x,y\in \Lambda}\; \sup_{|\alpha|_1 + |\beta|_1 \le p_\Phi}L^{(|\alpha|_1+ |\beta|_1)j}|\nabla_x^{\alpha}\nabla_y^{\beta}C_{j;x,y}|.
V ( \theta ) = \frac{4 \Omega^{2}}{{( 1 - v^{2})}^{2}}\frac{\sin^{4}\frac{\theta}{2}}{\cos^{2}\frac{\theta}{2}}[ \sin^{2}\frac{\theta}{2}- ( 1 - v^{2}) ] + \frac{3 \Omega^{2}}{2 ( 1 - v^{2})}.
\begin{aligned}[y_2^{\mu_2}] f_{k_2}(y_2)^{N-\nu}[y^{\mu_1+\mu_{1,2}}]f_{k_2}(y)^{\nu}&\le\! \frac{\gamma_{k_2}^2}{\sqrt{\mu_2(\mu_{1,2}+\mu_1)}}\frac{f_{k_2}(z_o)^N}{z_o^{M+1}}\!\left(\frac{M}{N}\!\right)^{M+1}\!\!\!\left(\!\frac{N-\nu}{\mu_2}\!\right)^{\mu_2}\!\!\left(\frac{\nu}{\mu_1+\mu_{1,2}}\!\right)^{\mu_1+\mu_{1,2}}\!\!.\end{aligned}
T\overline\partial^{*}_b=\overline\partial^{*}_bT\ \ \mbox{on$\Omega^{0,\bullet}(X,E)$}, &\overline\partial^{*}_{b,m}=\overline\partial^{*}_b|_{\Omega^{0,\bullet}_m}:\Omega^{0,\bullet}_m(X,E)\rightarrow\Omega^{0,\bullet}_m(X,E),\ \ \forall m\in\mathbb{Z}.
\mathbf M_{ij}-\frac{(\lambda_{n+2}-\lambda_i)(\lambda_{n+3}-\lambda_i)(\alpha_1-\lambda_1)}{(\lambda_{n+2}-\lambda_1)(\lambda_{n+3}-\lambda_1)(\alpha_1-\lambda_i)}\mathbf M_{1j}=\frac{(\lambda_1-\lambda_i)}{(\alpha_1-\lambda_i)}N_{ij}.
\varepsilon^{( 1 )}( \vec{x}) = \frac{g_{\varepsilon}\lambda^{2}}{4 \pi c \tau_{S}}\frac{\cos ( | \vec{x}- \vec{x}_{S}| / \lambda )}{| \vec{x}- \vec{x}_{S}|}\sum_{l m_{l}}c_{l m_{l}}Y_{l m_{l}}( \theta , \phi ) .
c_1(\tau(z)) = Q_1(1_{\tau(z)}) - \sum_{w \in X_1}b_1(\tau(z),w) &= \beta \left(h(z)^2 Q_2(1_z) - \sum_{v \in X_2}h(z)h(v) b_2(z,v) \right) &= \beta h(z) \left(\sum_{v \in X_2}b(z,v)h(z) + c_2(z)h(z) - \sum_{v \in X_2}h(v) b_2(z,v) \right) &= \beta h(z) \mathcal{L}_2 h (z).
P_{\geq k-m}( P_{[k-m,k+m)}A^j \partial_j P_k \phi) = \Q_{< k -Cm}P_{\geq k-m}( Q_{<k-Cm}P_{[k-m,k+m)}A^j \partial_j Q_{<k-Cm}P_k \phi) \ + \mathcal M_{A,large}^{m,main}P_k \phi
\|(\prod_{i=1}^{10}\sum_{\atop{l(\tau)=N^{-1/4}}}|E_{\tau}g_i|^2)^{1/20}\|_{L^{p}(w_{B})}\lesssim_{\nu,\epsilon,p}N^{-\frac{15}{4p}+\epsilon}(\prod_{i=1}^{10}\sum_{\atop{l(\tau)=N^{-1/4}}}\|E_{\tau}g_i\|_{L^{2p/5}(w_{B})}^2)^{\frac1{20}}.
\mathcal A_{11}={\partial (f_3(c(t_3)), f_3(c(t_4)), \cdots, f_3(c(t_{5d})))\over\partial (\theta_2^0, \cdots, \hat \theta_1^1, \cdots, \theta_d^2, \gamma_1, \cdots, \gamma_{2d})}, ( \hat{\cdot}:=omit)
A_s =a_{2s,1}+a_{2s,-1}+a_{2s-1}, A'_r =a'_{2r,1}+a'_{2r,-1}+a'_{2r-1}, B_s =a_{2s,1}+a_{2s,-1}+a_{2s+1}, B'_r =a'_{2r,1}+a'_{2r,-1}+a'_{2r+1}, C_s =q^{-s}(a_{2s,1}-a_{2s,-1}), C'_r =\frac{\alpha_{-1}}{\beta_r}a'_{2r,1}-\frac{\alpha_{1}}{\beta_r}a'_{2r,-1},
G(\theta)= \mathbb{E}_{\Phi|\Phi(\mathcal{A})>0}\left\{\left(\sum_{m=1}^{M}C_{M}^m(-1)^{m+1}e^{-m\eta\theta r_1^{\alpha}/\rho}\right.\right. \left.\left.\prod\limits_{x_{i}\in\Phi\backslash\{x_1\}\cap{\mathcal{A}}}\frac{1}{\left(1+\frac{m\eta\theta r_1^{\alpha}}{Mr_i^{\alpha}}\right)^M}\right)^b\right\},
D_\lambda(p_{W_{T}, \hat W_{T\times \{d\}}}\|p_{W_{T}}s_{\hat W_{T\times \{d\}}}) \le \sum_{k=1}^n D_\lambda( p_{Y_{T^c,k}|X_{\mathcal{I},k}}^{(\lambda, T)}\|p_{Y_{T^c,k}|X_{T^c,k}}^{(\lambda, T)}|p_{X_{\mathcal{I},k}}^{(\lambda, T)}).
Var(\sum_{i=1}^{n}X_{i})=E(|\sum_{i=1}^{n}X_{i}|^{2})=\sum_{i=1}^{n}\sum_{j=1}^{n}E(X_{i}\overline{X}_{j})=\sum_{i=1}^{n}E(|X_{i}|^{2})=\sum_{i=1}^{n}Var(X_{i}).
0 = F_{u}^{\prime \prime}+ \frac{2}{r}F_{u}^{\prime}\left[ 1 + r L_{-}^{\prime}( k r ) \right] - \frac{s^{\prime}}{s + \frac{\omega}{m}}\left[ F_{u}^{\prime}+ F_{u}L_{-}^{\prime}( k r ) \right] + W ( r , - m ) F_{u}- \left[ K , F_{u}\right]
b_{4s+1}= b_{2s}+b_{2s+1}= -\frac{1}{k}b_s + b_{2s+1}\geq b_{2s+1}-\frac{1}{4}|b_s| > b_{2s+1}-\frac{1}{2}b_{2s+1}= \frac{1}{2}b_{2s+1}.
{\mathbb{T}}_{\vec{b}}^{j}(\mathbf{n}_{1},\mathbf{n}_{2},\mathbf{n}_{3})=\left( \frac{n+1}{4\pi}\right)^{2}A_{W}^{n}(\mathbf{n}_{1},\mathbf{n}_{2},\mathbf{n}_{3})\exp{\{i\Phi_{W}^{n}(\mathbf{n}_{1},\mathbf{n}_{2},\mathbf{n}_{3})\}}
E=E_{c}+\frac{\hbar^{2}(q_{1}\pi/d_{1})^{2}}{2m_{c}}+\frac{\hbar^{2}(q_{2}\pi/d_{2})^{2}}{2m_{c}}+\frac{\hbar^{2}(q_{3}\pi/d_{3})^{2}}{2m_{c}}
\overrightarrow{\not \! \! D}_{R}\equiv e_{a}^{\mu}\gamma^{a}\left( \vec{\partial}_{\mu}+ \frac{1}{2}\omega_{\mu}^{c d}\sigma_{c d}+ i g^{\prime}B_{\mu}\right) ; \quad \overleftarrow{\not \! \! D}_{R}\equiv \left( \overleftarrow{\partial}_{\mu}- \frac{1}{2}\omega_{\mu}^{c d}\sigma_{c d}- i g^{\prime}B_{\mu}\right) \gamma^{a}e_{a}^{\mu};
\sum_{k=0}^{\frac{p^r-1}{2}}\frac{(\frac{1}{2})_k^3}{k!^3}(3k+1)2^{2k}\equiv p^r \pmod{p^{r+2}}, begin{align*}5pt]\sum_{k=0}^{p^r-1}\frac{(\frac{1}{2})_k^3}{k!^3}(3k+1)2^{2k}\equiv p^r \pmod{p^{r+2}}.
q^{-n}\big(-S_n(1)+S_{n-1}(1)+ q^n (q;q^2)_n\big) &=q^{-n+1}\big(-S_{n-1}(1)+S_{n-2}(1)+q^{n-1}(q;q^2)_{n-1}\big) & -q^{n-1}\big(S_{n-2}(1)-q S_{n-1}(1)+q^{n-1}(1+q)(q;q^2)_{n-1}\big) +q^{2n-2}(q;q^2)_{n-1}.
e^{2 \beta z_{+}^{\prime}}= e^{- 2 \beta z_{-}^{\prime}}= 1 + \frac{2 \beta^{2}}{\chi^{2}}+ \frac{2 \beta}{\chi^{2}}\left( \chi^{2}+ \beta^{2}\right)^{1 / 2}\geq 1
([\begin{matrix}y_{0}\\y_{1}\\y_{2}\\ \cdot\cdot\cdot\\y_{m-1}\end{matrix}],[\begin{matrix}y_{m}\\y_{m+1}\\y_{m+2}\\ \cdot\cdot\cdot\\y_{2m-1}\end{matrix}],...,[\begin{matrix}y_{n-m}\\y_{n-m+1}\\y_{n-m+2}\\ \cdot\cdot\cdot\\y_{n-1}\end{matrix}])
{\cal L}^{( 0 )}= \pi_{i j}\dot{A}^{i j}+{\frac{1}{2}}\pi_{i j}\pi^{i j}+ 2 \partial_{i}A_{j 0}\pi^{i j}-{\frac{1}{2}}\partial_{i}A_{j k}\partial^{i}A^{j k}+ \partial_{i}A_{j k}\partial^{j}A^{i k},
(\hat\gamma_0-\hat\gamma)\rho_0 = -\hat\gamma(\rho_0-\rho) + (\rho-\rho_0) \frac{\int \frac{t^2}{(\hat\gamma\rho+(1-\rho)t)(\hat\gamma_0\rho_0+(1-\rho_0)t)}\nu(dt)}{\int \frac{t}{(\hat\gamma\rho+(1-\rho)t)(\hat\gamma_0\rho_0+(1-\rho_0)t)}\nu(dt)}.
\hat{g}(x,a,b) = \frac{I_a^{(q,r)}(x)}{I_a^{(q,r)}(b)}+\Big( J_a^{(q,r)}(x)- J_a^{(q,r)}(b)\frac{I^{(q,r)}_a(x)}{I^{(q,r)}_a(b)}\Big) \frac{\hat{g}(0,a,b)}{Z^{(q)}(-a)}.
(-1)^k \displaystyle \sum_{l=0}^{k}\displaystyle \sum_{r=0}^{k}(-1)^{r+1}\genfrac{\{}{\}}{0pt}{}{k}{r+1}\begin{bmatrix}r+1 l\end{bmatrix}=(-1)^k\displaystyle \sum_{l=0}^{k}(-1)^l \delta_{l,k}&=(-1)^k \cdot (-1)^k &=1.
b(v,w;\xi) = \int_{D\times I}\frac{\partial v}{\partial t}w_1 + \int_{D\times I}\kappa(\cdot,\xi) \nabla v \cdot \nabla w_1 + \int_{D}v(\cdot,0) w_2 , &\langle f(\xi) , w \rangle = \int_{D\times I}g(\cdot,\cdot,\xi) w_1 + \int_D u_0(\cdot,\xi) w_2.
\int_{\Re}W(x)\bigl(f'(y)\bigr)^2\bigl(s'(x)\bigr)^2\, \mu(dx) &\quad= \int_{\Re}\bigl(f'\bigl(s(x)\bigr)\bigr)^2 \biggl({\hat F_\mu(y)\over p_\mu(y)}\biggr)^2\log{1\over2\hat F_\mu(y)}p_\mu\bigl(s(y)\bigr)\,{p_\mu(y)\over p_\mu(s(y))}dy &\quad=\int_{\Re}W(y) \bigl(f'(y)\bigr)^2\, \mu(dy),
c_i(tu,v) = c_i(tu,v^{2k+1})c_i(tu,v^{2k})^{-1}\:\:\mbox{(By (iii))}= c_i(t,v^{2k+1})c_i(u,v^{2k+1})c_i(t,v^{2k})^{-1}c_i(u,v^{2k})^{-1}\:\:\mbox{(By (L6) and (v))}= c_i(t,v)c_i(t,v^{2k})c_i(u,v)c_i(u,v^{2k})c_i(t,v^{2k})^{-1}c_i(u,v^{2k})^{-1}\:\:\mbox{(By (iii))}= c_i(t,v)c_i(u,v)
H^{1}({\cal S},{\cal O}_{{\cal S}}(-1) \oplus\bigoplus_{i=3}^{7}{\cal O}_{{\cal S}}(-i))=H^{0}({\cal S},{\cal O}_{{\cal S}}(-1) \oplus\bigoplus_{i=3}^{7}{\cal O}_{{\cal S}}(-2+i))^{*}.
\mathbb{E}[\frac{\sum_{j=1}^{d}(\frac{CS_j}{l_j})}{d}] =\frac{1}{d}(\sum_{j=1}^{d}(\mathbb{E}[\frac{CS_j}{l_j}])) &=\frac{1}{d}\sum_{j=1}^{d}(\frac{\sum_{i=1}^{l_j}\mathbb{E}[\sigma^{X}_{j_i}]}{l_j}) &=\mathbb{E}[\sigma^{X}_1]=\frac{n+1}{2}.
\sup_{0\leq t \leq T}\|\nabla \varphi(t,\cdot)\|_{q}\leq 2\|\nabla\varphi_o\|_{q}&+ C_{n}\Big(1+\sup_{0\leq s \leq t}\|f(s,\cdot)\|_{1}\Big)\sup_{0\leq s \leq T}\|f(s,\cdot)\|^{\frac{\theta}{n}\frac{q(n-1)-n}{\theta q -2}}_{q\theta/2}\,.
\mathbf{B}:=\bigcup_{j\in J}\bigcup_{w\in X_j}\{\mathrm{ev}_{\mathtt{v}}\circ \partial_w\circ \xi_j\}\qquad\underline{\mathbf{B}}:=\bigcup_{j\in J}\bigcup_{w\in G_{\mathtt{v}}}\{\mathrm{ev}_{\mathtt{v}}\circ \partial_w\circ \xi_j\}.
\frac{1}{b_{n}}\sum_{k=2}^{n}\mathrm{cov}(K(\frac{x_{j}-X_{0}}{b_{n}}),K(\frac{x_{p}-X_{k}}{b_{n}})) \ll\sum_{k=2}^{m_{n}}b_{n}+\sum_{k=m_{n}+1}^{n}\frac{1}{b_{n}^{3}}\eta_{k}& \ll m_{n}b_{n}+\frac{1}{b_{n}^{3}m_{n}^{3}l\left( m_{n}\right)}.
D^{l}e^{\frac{\alpha}{p}M_{s,t}}=\left(\frac{\alpha}{p}\right)^{l}e^{\frac{\alpha}{p}M_{s,t}}DM_{s,t}\otimes\cdots\otimes DM_{s,t}\in L^{p}(\boldsymbol{W};\mathcal{H}^{\otimes l})
\chi = \left( 1 2 8 \pi^{2}\right)^{- 1}\int d^{4}x ~ g \varepsilon_{\mu \nu}^{\tau \omega}R_{\tau \omega \alpha \beta}\varepsilon_{\rho \sigma}^{\alpha \beta}R^{\mu \nu \rho \sigma}~ + ~ b o u n d a r y ~ t e r m s
Z = \int [{\cal D}Q_{\mu}\,{\cal D}\overline{{{c}}}\,{\cal D}c ] \, e^{- \int d^{3}x \left({\cal L}+ J_{\mu}Q_{\mu}+ \overline{{{J}}}_{c}c + \overline{{{c}}}J_{\overline{{c}}}\right)}\, ,
\sum_{\ell_{q-1}<j <\ell_q}(-1)^{p(\eta,j)-1}i_{\eta \cup j}= U\left(\prod_{r=m+1}^{q-1}{D_{\ell_{r}-1}}\prod_{r=q+1}^{M}{D_{\ell_{r-1}}}\right)\left(-\sum_{\ell_{q-1}<j<\ell_q}{x_{j}}\right) &= U\left(\prod_{r=m+1}^{q-1}{D_{\ell_{r}-1}}\prod_{r=q+1}^{M}{D_{\ell_{r-1}}}\right)\left(D_{\ell_{q-1}}-D_{\ell_q-1}\right) &= U(B(q-1)-B(q)).
\begin{aligned}y_jy_i ={\lambda}_{ji}y_iy_j 1\leq i<j\leq n y_jx_i ={\lambda}_{ij}x_iy_j 1\leq i<j\leq n x_jy_i ={q}_i{\lambda}_{ij}y_ix_j 1\leq i<j\leq n x_jx_i ={q}_{i}^{-1}{\lambda}_{ij}^{-1}x_ix_j 1\leq i<j\leq n x_iy_i-{q}_iy_ix_i =1+\sum_{k=1}^{i-1}(q_k-1)y_kx_k \qquad 1\leq i\leq n.\end{aligned}
\begin{aligned}u (\bar{x}) = \frac{1}{2}\sum_{\beta=1}^{n-1}b_\beta \bar{x}_\beta^2 + + \frac{1}{6}\sum_{\xi, \beta, \gamma}\varphi_{\xi \beta \gamma}(0) \bar{x}_\xi \bar{x}_\beta \bar{x}_\gamma + O (|\bar{x}'|^4) & \leq u (x_0) + \frac{1}{2}\sum_{\beta=1}^{\alpha}b_\beta \bar{x}_\beta^2 + C \delta^3 b_\alpha^2. \end{aligned}
\mathfrak{R}(f,\sigma,\{k_j\},L)= &\left\{\{R_i\}\in[0,L-1]^n\left|\begin{array}{l}R_1+R_4\equiv R_2+R_3\bmod L(R_1+R_4,L)=1, R_1+R_4=1\mathbb{Q}(\zeta_8)\cap\mathbb{Q}(\zeta_L)\end{array}\right.\right\}.
{\cal H}= \frac{1}{2}( \vec{E}^a \vec{E}^a + \vec{B}^a \vec{B}^a)+ \frac{1}{2}( \vec{\cal E}^{\,2}+ \vec{\cal B}^{\,2})+ \pi^{\dagger}\pi + ( \vec{D}\varphi )^{\dagger}( \vec{D}\varphi ) + V ( \varphi^{\dagger}\varphi )
\Delta z\approx\frac{\partial z}{\partial x_{1}}\Delta x_{1}+\frac{\partial z}{\partial x_{2}}\Delta x_{2}+\frac{\partial z}{\partial x_{3}}\Delta x_{3}+\cdot\cdot\cdot=\sum_{i=1}^{p}\frac{\partial z}{\partial x_{i}}\Delta x_{i}Eq(6)
|\partial_{r}N_{2}(f_{v_{5}})(t,r)| \leq \frac{C}{r^{3}t^{3}\log^{4N+7b-2}(t)}+ \frac{C \log(r)}{r^{3}|t-r| t^{3/2}\log^{4b-1+\frac{5N}{2}}(t)}+ \frac{C \log^{2}(r)}{r^{2}t^{3/2}(t-r)^{2}\log^{3b-1+\frac{5N}{2}}(t)}&+\frac{C \log^{3}(r)}{r^{2}(t-r)^{4}},t-\sqrt{t}> r > \frac{t}{2}r > t+\sqrt{t}