harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\cal{Z}}^{PF}(X_1, X_2 ; p) =\frac{1}{(1-X_1)}\frac{1}{(1-X_2)}\left[\frac{(1-(X_1 X_2)^{p+2})}{(1-X_1 X_2)}- \frac{({X_1}^{p+2}-{X_2}^{p+2})}{(X_1 - X_2)}\right]~~~~.
Q = \left( \begin{array}{c c c c c c}{{0}}&{{0}}&{{0}}&{{0}}&{{0}}&{{- 1}}\\{{1}}&{{0}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{1}}&{{0}}&{{0}}&{{0}}&{{- 1}}\\{{0}}&{{0}}&{{1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{1}}&{{0}}&{{- 1}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{1}}&{{0}}\\ \end{array}\right) .
\mid{P}_{0}\rangle \:=\:{\mathbf O^{-1}}\mid \tilde{P}_{0}\rangle \:=\:\exp \left({\cal W}^{\mu\sigma}_{\nu\rho}{\mathbf a}^{\nu}{\mathbf b}_{\mu}{\mathbf a}^{\rho}{\mathbf b}_{\sigma}\right)\prod_{\alpha =0}^{3}\,{\mathbf a}^{\dagger}_{\alpha}\prod_{\beta =0}^{3}\,{\mathbf b}^{\dagger\beta}\, \mid 0 \rangle.
F_{k}( x^{\omega_{1}}\Phi^{1}, x^{\omega_{2}}\Phi^{2}, . . . , x^{\omega_{N}}\Phi^{N}) = x^{1 - \rho_{k}}F_{k}( \Phi^{1}, \Phi^{2}, . . . , \Phi^{N}) , \quad 1 \le k \le M ,
\psi(h_1) =\psi(h_3)=\psi(h_2)=\psi(h_4)=\frac{n+1}{4}e \, n\equiv 3 \pmod{4}\,, \psi(h_1) =\psi(h_3)=\psi(h_2)=\psi(h_4)=\frac{3n+1}{4}e n\equiv 1 \pmod{4}\,.
P_{2n}(x;a,a,1,1;q) =\frac{p_n(x^2;q^{-1},a^2;q^2)}{p_n((qa)^{-2};q^{-1},a^2;q^2)}\,, P_{2n+1}(x;a,a,1,1;q) =\frac{qax\,p_n(x^2;q,a^2;q^2)}{p_n((qa)^{-2};q,a^2;q^2)}\,.
\mathbb{E}(\|\mathcal{L}_L^{\lambda}f^{\epsilon}-f\|_2)\leq c\tau_3 V^{1/2}G\sqrt{\log{N}}+\left[(1+\tau_3)V^{1/2}C\|f\|_{k,\zeta}\right]L^{-k-\zeta}&+\left[V\left(\|f\|_{\infty}+C\|f\|_{k,\zeta}L^{-k-\zeta}\right)^2-\chi\right]^{1/2},
{M}_{6}= \sum_{i=1}^{n}C^i_n g_{i}\eta_2 \partial^{n -i}_x{v}+\sum_{i=1}^{n}C^i_n g_{i}\partial^{n+1 -i}_x{u}+ \sum_{i=1}^{n}C^i_n \partial_y g_{i}\partial^{n -i}_x{v}& + \sum_{i=1}^{n}\partial_y^{-1}g_{i}\left( C_n^i\partial^{n -i}_x{v}\partial_y \eta_2 + C_n^i \partial^{n+1 -i}_x{u}\eta_2\right).
(2n+2)T(r)\geqslant 2\sum_{u=1}^3\sum_{t=1}^{2n+2}N^{(n)}_{(f^u,H_t),\leqslant k_t}(r)+(n-2)\sum_{t=1}^{2n+2}N^{(1)}_{(f,H_t),\leqslant k_t}(r)+\sum_{t=1}^{2n+2}N(r,\nu_t) & -(n+1)\sum_{u=1}^3\sum_{t=1}^{2n+2}N_{(f^u,H_i),> k_i}+o(T(r)).
h_{H}(f^{n}(x)) =\sum_{k=0}^{n-1}\delta_{f}^{n-1-k}\bigl(h_{H}(f^{k+1}(P))-\delta_{f}h_{H}(f^{k}(P))\bigr)+\delta_{f}^{n}h_{H}(P) &=\sum_{k=0}^{n-1}\delta_{f}^{n-1-k}B(f^{k}(P))+\delta_{f}^{n}h_{H}(P).
\hat{I}[a_{cl}+\delta a] \approx \hat{I}_{cl}+ \frac{1}{2}\int_0^T dt \delta a(t)\left[-2 \Lambda a_{cl}- 2\ddot{a}_{cl}- 2\dot{a}_{cl}\frac{d}{dt}- 2 a_{cl}\frac{d^2}{dt^2}\right] \delta a(t)
\begin{array}{lcl}\Psi_{n}\big((r_0,...,\hat{r_j},...,r_{n})\big) = \psi_{n}\circ Q_{n}\big((r_0,...,\hat{r_j},...,r_{n})\big) &= \psi_{n}(r_0dr_1\wedge...\wedge\hat{dr_j}\wedge...\wedge dr_n) &= \psi_{n}(r_0dr_k\wedge d\bar{r}')^{(-1)^{k-2}}.\end{array}
\frac{1}{2m}(\frac{dS_{z}}{dz})^{2}+U_{z}(z)+\frac{1}{2m(\sigma^{2}+\tau^{2})}[(\frac{dS_{\sigma}}{d\sigma})^{2}+(\frac{dS_{\tau}}{d\tau})^{2}+2mU_{\sigma}(\sigma)+2mU_{\tau}(\tau)]=E.
dh(t,x,z) =y(t,x,z)[-\{L^*_{\pi(t)}p(t,x,z)+xq(t,x,z)\}dt + q(t,x,z) dG(t)] &+p(t,x,z)[L^*_{\pi(t)}y(t,x,z)dt+xy(t,x,z)dG(t)] &+xy(t,z)q(t,x,z)d\langle G \rangle_t &=y(t,x,z)q(t,x,z) dG(t)+p(t,x,z)xy(t,x,z)dG(t) &=dF(t,x,z)+xh(t,x,z)dG(t)
\mu^* =\mu^{\rm N}-\varkappa^{\rm N},\qquad\, \varkappa=2\varkappa^{\rm N},\qquad\lambda=\lambda^{\rm N}, \beta =\gamma^{\rm N}-\beta^{\rm N}, \ \gamma =\gamma^{\rm N}+\beta^{\rm N}, \,\qquad\alpha =\alpha^{\rm N}\,.
\tilde U_\star=\frac{2\delta^2 G^2+c^2+m(c^2+ 2G^2(1+\delta^2))\alpha+2m^2G^2\alpha^2}{(m^2-\tilde M \delta^2)-m(\tilde M(1+\delta^2)-2m^2) \alpha-\tilde M m^2\alpha^2}
f-f_n = \sum_{a_1,b_1, \ldots a_k,b_k=-n}^n \sum_{i=1}^k\Big(g_i\otimes h_i - \frac{a_i b_i}{n^2}g\otimes h\Big) 1_{A_{a_1\ldots a_kn}\times B_{b_1\ldots b_kn}},
\mathcal{Z}_p:=\left\{f=f(t,\xi):\t^{1-\frac{1}{p}-\gamma}f\in C_b([0,\infty);L^p(\mathbb{R}^2)),\t^{\frac{3}{2}-\frac{1}{p}-\gamma}\partial_j f\in C_b([0,\infty);L^p(\mathbb{R}^2)),\j=1,2\right\}.
m \leq 1 + \sum_{y\in\mathcal{Y}}\sum_{\ell=1}^{J_y}(b^y_\ell - 1) \leq 1 + \sum_{y\in\mathcal{Y}}\sum_{\ell=1}^{J_y}\left(\frac{3}{4}index(\Sigma^y_l)+\frac{1}{4}\right) \leq 1+\frac{3}{4}j+\frac{1}{4}\sum_{y\in\mathcal{Y}}J_y\leq 1+j.
\left\{{\sum\limits_{i = 1}^n{{a_i}^2}}\right\}\left\{{\sum\limits_{i =1}^n{{b_i}^2}}\right\}-{\left\{{\sum\limits_{i = 1}^n{{a_i}{b_i}}}\right\}^2}\le \frac{n^2}{4}{\left\{{M_1}{M_2}-{m_1}{m_2}\right\}^2}
= \frac{1}{4}[L_{0},L_{1}] + \frac{1}{4}(L_{0}^{2}g -L_{0}gL_{0}) + \frac{1}{4}(L_{1}gL_{1}- L_{1}^{2}g) + \frac{1}{4}(L_{1}gL_{0}g -L_{0}gL_{1}g)=
A^{2}= \int x^{2}(\partial_x^6u)^2dx &= \int u\partial_x^6(x^2\partial_x^6u)dx &= \int x^{2}u\partial_x^{12}udx+12\int xu\partial_x^{11}udx+30\int u\partial_x^{10}udx &\lesssim \|x^{2}u\|_2\|\partial_x^{12}u\|_2+ \|xu\|_2\|\partial_x^{11}u\|_2+ \|u\|_2\|\partial_x^{10}u\|_2.
\prod_{i=1}^N(1-zq^{i-1}) = \sum_{j=0}^N{N\brack j}_q(-1)^j z^j q^{j(j-1)/2}; \frac{1}{\prod_{i=1}^N(1-zq^{i-1})}= \sum_{j=0}^{\infty}{N+j-1\brack j}_q z^j.
V(\nu_1,\nu_2,{\bar \nu_1},{\bar \nu_2}) = k\cdot \log{\langle{\nu_1},{\nu_2}\vert{\bar \nu_1},{\bar{\nu_2}}\rangle}= k\cdot \log( 1+{\nu_1}{\bar{\nu_1}}+{\nu_2}{\bar \nu_2}) = k\cdot \log{\langle \nu \vert \nu \rangle}
T = - \frac{1}{2}( \partial X )^{2}- \frac{1}{2}( \partial \phi )^{2}- \frac{1}{\sqrt{2 k}}\partial^{2}\phi - \frac{1}{2}( \partial H^{\prime})^{2}+ \left\{Q_{U ( 1 )}, ~ \frac{1}{\sqrt{2 ( k + 2 )}}\eta ( - \partial u + i \partial v^{\prime}) \right\}.
A(t)U(t,0)x = -i(\Delta +b(t,\xi) \cdot \nabla+c(t,\xi))U(t,0)x = -i\Delta U(t,0)x-ib(t,\xi)\cdot \nabla U(t,0)x-ic(t,\xi)U(t,0)x = -i\Delta U(t,0)x-2i(\nabla W)(t,\xi)\cdot \nabla U(t,0)x-i\sum_{j=1}^d(\partial_jW(t,\xi))^2U(t,0)x & -i(\Delta W)(t,\xi)U(t,0)x-\mu(\xi)U(t,0)x-\widetilde{\mu}(\xi)U(t,0)x.
\| \hat u^3 - \hat v^3 \|_{L^2}= \| (\hat u - \hat v)( \hat u^2 + \hat v^2 + \hat u \hat v) \|_{L^2}\lesssim \| \hat u - \hat v \|_{L^{q_1}}( \| \hat u\|^2_{L^{q_2}}+ \| \hat v\|^2_{L^{q_2}}+\| \hat u\|_{L^{q_2}}\| \hat v\|_{L^{q_2}}) & \lesssim ( \| \hat u\|_{H^{m_d-1}}^2 + \| \hat v\|_{H^{m_d-1}}^2 ) \|\hat u - \hat v\|_{H^{m_d-1}}.
n^\beta\exp\{-9\alpha^{-2(i+1)}f(n,p)^2\}\geq \exp\{\beta\log n-9\alpha^{-2(i+1)}\sqrt{\log n}\}&= \exp\{\beta\log n(1-9\alpha^{-2(i+1)}(\log^{-1/2}n))\}&= \omega(1),
ds^2 ={r^2 \over L^2}\left[ - \left (1-{r_0^4\over r^4}\right )dt^2 + d\vec{x}^2 \right] +{L^2 \over r^2}\left ( 1-{r_0^4\over r^4}\right )^{-1}dr^2 + L^2 d\Omega_5^2 \ ,
\frac{n-2}{n}\left(\frac{n-2}{n}\beta\lambda+\alpha\right) =\frac{n-2}{n}\left(\frac{(n-2)(n+2)}{n}\lambda +n+2-(n-2)\lambda\right) =\frac{n-2}{n}\left[ \left(\frac{n+2}{n}-1\right)(n-2)\lambda+n+2\right] =\frac{n-2}{n}\left[\frac{2(n-2)}{n}\lambda+n+2\right]\geq(n-2)\frac{n+2}{n}>n-2.
\big((1-\lambda)I+\lambda A^{\frac{1}{2}}B^{-1}A^{\frac{1}{2}}\big)^{\frac{1}{2}}\big((1-\lambda)I+\lambda A^{-\frac{1}{2}}BA^{-\frac{1}{2}})\big((1-\lambda)I+\lambda A^{\frac{1}{2}}B^{-1}A^{\frac{1}{2}}\big)^{\frac{1}{2}}-I &=\lambda(1-\lambda)(I-A^{-\frac{1}{2}}BA^{-\frac{1}{2}})A^{\frac{1}{2}}B^{-1}A^{\frac{1}{2}}(I-A^{-\frac{1}{2}}BA^{-\frac{1}{2}}).
{\bar{e}}_{\alpha}^{\;\; i}{\bar{e}}_{\beta}^{\;\; j}\left[{B}_{ij}^{\bar{g}'(2)}\right] ={\bar{e}}_{\alpha}^{\;\; i}{\bar{e}}_{\beta}^{\;\; j}\frac{\partial}{\partial\bar{g}^{ij}}[ \Gamma'^{(2)}] +{K}^{g'}_{\alpha \beta}
W [ J ] = \Gamma ( \phi_{\mathrm{c l .}}) + \int d^{d}x{\frac{\delta W [ J ]}{\delta J ( x )}}J ( x ) , \quad{\frac{\delta W [ J ]}{\delta J ( x )}}= \phi_{\mathrm{c l .}}= < \Omega | \phi | \Omega >_{J}\quad ,
\phi \equiv \frac{1}{3}\tan^{- 1}\left( \frac{2}{\sqrt{\frac{q^{2}- q_{-}^{2}}{q_{+}^{2}- q^{2}}}- \sqrt{\frac{q_{+}^{2}- q^{2}}{q^{2}- q_{-}^{2}}}}\right) \, .
\{\; \; , \; \; \}_{\alpha}= ( \{\; \; , \; \; \}_{\Lambda}, \; \{\; \; , \; \; \}_{W}) , \quad \left( I_{a}^{\alpha}, Q_{a}^{\alpha}, D^{\alpha}\right) = \left( ( I_{a}, Q_{a}, D_{0}) , ( \phi_{a}, q_{a}, D ) \right) . \nonumber
\{H^{\beta}(z^2), H^{\gamma}(w^2)\}=i_{z, w}\frac{1}{(z^2 -w^2)^2}- i_{w, z}\frac{1}{(w^2 -z^2)^2}; &\{H^{\beta}(z^2), H^{\beta}(w^2)\}= 0; \{H^{\gamma}(z^2), H^{\gamma}(w^2)\}= 0.
f(x_1, x_2, x_3)= \sum \limits_{j,k}\sum_{R=I\times J\times S\in \mathcal R^N_{\frak z}(j,k)}|R|\widetilde \psi_{j,k}(x_1, x_2, x_3,x_I,x_J,x_S) \ \psi_{j,k}*f(x_I, x_J, x_S),
\delta S^{cd}_{ab}\left( \vartheta \right) =-ig \frac{F_{\bar{c}\bar{d}ab}^{\Psi}\left( i\pi ,\, \vartheta +i\pi ,\, 0,\, \vartheta \right)}{m_{a}m_{b}\sinh \vartheta}\quad ,\quad \vartheta =\vartheta_{a}-\vartheta_{b}\: .
\Delta u_0' =-4\sum\limits_{j=1}^{N_1}{\dfrac{\mu}{(\mu+|x-p_j|^2)^2}}+4\pi\sum\limits_{j=1}^{N_1}{\delta_{p_j}(x)}, \Delta u_0'' =-4\sum\limits_{j=1}^{N_1}{\dfrac{\mu}{(\mu+|x-q_j|^2)^2}}+4\pi\sum\limits_{j=1}^{N_2}{\delta_{q_j}(x)}.
\left( \frac{k_{\perp}^{\prime}}{k^{' +}}e^{- i \lambda \varphi^{\prime}}+ \frac{k_{\perp}}{k^{+}}e^{+ i \lambda \varphi}\right) \delta_{- \lambda^{\prime}}^{\lambda}- m \lambda \left( \frac{1}{k^{' +}}+ \frac{1}{k^{+}}\right) \delta_{\lambda^{\prime}}^{\lambda}
V_{z \ll b , \, x \ll \sqrt{Q_{p}b}}\to E -{\frac{m_{f}}{2 Q_{p}Q_{F}}}z ( x^{2}+ 4 Q_{p}z )^{\frac{n}{2}}+{\frac{m_{f}J_{\perp}^{2}}{2 E^{2}Q_{p}^{2}Q_{F}^{2}}}( x^{2}+ 4 Q_{p}z )^{n},
\eta_{e}:= 2 \sigma_e \{h^{-1}\}=\begin{cases}2 \sigma_e h_i^{-1}e \in \Gamma_D, e \subset \Omega_i, \sigma_e \Big( h_i^{-1}+ h_j^{-1}\Big) e \in \Gamma_I, e \subset \Omega_i \cap \Omega_j.\end{cases}
\psi_{x_1}=\psi_{x_2}=\psi_{x_6}=\psi_{x_7}=(x_4x_5)(x_9x_{10}), \psi_{x_3}=\psi_{x_8}=(x_1x_4x_2x_{10})(x_5x_6x_9x_7), \psi_{x_4}=\psi_{x_5}=\psi_{x_9}=\psi_{x_{10}}=(x_1x_7)(x_2x_6).
H = - i{\alpha}^{j}\frac{\partial}{\partial \xi^{j}}- i{\alpha}^{3}\frac{\partial}{\partial s}+ \beta m - i{\alpha}^{3}\Bigl ( \frac{\vec{\kappa}\cdot \vec{\xi}}{\Delta}\frac{\partial}{\partial s}+ \frac{1}{2 \Delta}\frac{\partial ( \vec{\kappa}\cdot \vec{\xi})}{\partial s}\Bigr ) \ .
\alpha^* =1+\lim_{z\rightarrow 0}z\frac{\Phi(z)}{\phi(z)}\frac{1-\Phi(z)}{1-2\Phi(z)}=1+\frac{\sqrt{2\pi}}{4}\lim_{z\rightarrow 0}\frac{z}{1-2\Phi(z)}&=1+\frac{\sqrt{2\pi}}{4}\lim_{z\rightarrow 0}\frac{1}{-2\phi(z)}=1-\frac{\pi}{4}\approx0.2146.
\begin{array}{l l}{I_{( a )}+ I_{( b )}+ I_{( c )}+ I_{( d )}}&{\sim \left[ G_{+ +}^{( 0 )}( k + p_{2}) - G_{- +}^{( 0 )}( k + p_{2}) \right] G_{R}( k - p_{1})}\\{{}}&{= G_{A}( k + p_{2}) G_{R}( k - p_{1}) .}\\ \end{array}
H_{0}^{\prime}= \frac{1}{2}\int d^{3}x \left( B_{a}^{i j}F_{i j}^{a}- A_{i}^{a}A_{a}^{i}+ \left( \partial^{i}B_{0 i}^{a}\right) \left( \partial_{j}B_{a}^{0 j}\right) \right) \equiv \int d^{3}x \, h_{0}^{\prime},
\sup_{V\in\mathcal{V}}\left\{\mathcal{E}_f(V)\ :\V\ge0,\ \int_\Omega|V|^p\,dx=1\right\}\le\sup_{V\in\mathcal{V}}\left\{\mathcal{E}_f(V)\ :\ \int_\Omega|V|^p\,dx\le 1\right\}.
L = \pi^{ij}\dot{g}_{ij}- N^{\mu}T_{\mu}+ P_{\mu}\dot{N}^{\mu}+ \bar{\tilde{C}}_{\mu}\dot{\tilde{C}}^{\mu}+ \bar{C}_{\mu}\{\dot{C}^{\mu}-U^{\mu}_{\alpha\beta}N^{\alpha}C^{\beta}-\tilde{C}^{\mu}\}.
H^{\epsilon}( z , \bar{z}) = \lim_{| z_{3}\bar{z_{3}}| \rightarrow \infty}| z_{3}\bar{z}_{3}|^{2 \Delta_{\alpha_{3}}}\langle \phi_{\alpha_{3}}\left( z_{3}, \bar{z}_{3}\right) \psi_{\alpha_{2}}\left( 1 \right) S_{0}R_{\alpha}^{\epsilon}\left( z , \bar{z}\right) R_{\alpha_{1}}^{\epsilon}\left( 0 \right) \rangle ,
\begin{cases}\ \ \Delta w+k^{2}(1+V)w=0 \mbox{$\mathrm{in}\ \ \Omega$}, \ \ \Delta v+k^{2}v=0 \mbox{$\mathrm{in}\ \ \Omega$}, \ \w=v,\ \partial_{\nu}w=\partial_{\nu}v+\eta v \mbox{$\mathrm{on}\ \ \Gamma$}, \end{cases}
(S_\lambda f^{m,v})(x) \overline{f^{m',v'}}(x)=\sum_{\mu\in D_v^J}c^{m,v}_\mu \rho(\Lambda)^{d(\lambda)/2}\Theta_{\lambda\mu}(x)\sum_{\mu'\in D_{v'}^J}\overline{c^{m',v'}_{\mu'}}\Theta_{\mu'}(x).
x_1 x_2 = x_2 x_1 + x_1^2, x_3 x_1 = x_1 x_3 + x_1^2 + 3 x_2^2, x_3 x_2 = x_2 x_3 - 3 x_2^2 - 2 x_1 x_3 - 2 x_1 x_2.
\psi ( 2 )_{\beta = \gamma = \delta = 0}= i \int \frac{d^{4}x_{1}d^{4}p^{\prime}}{( 2 \pi )^{4}}e^{- i p_{\mu}^{\prime}x_{2}^{\mu}}\; \frac{( \pm m ){u_{a}^{+}}^{\prime}{{u_{a}^{+}}^{\prime}}^{\dagger}\gamma^{4}}{{p^{4 \; \prime}}^{2}-{( m e^{{w_{1}}^{\prime}}\cosh w_{2}^{\prime})}^{2}+ i \epsilon}e^{i p_{\nu}^{\prime}x_{1}^{\nu}}N_{\sigma}\gamma^{\sigma}\psi ( 1 ) ,
\partial_r ={x (x - a) \over y a}e_{x y}-{x^2 r' \over a y r}e_{x y}' +{y \over a}e_{x y}-{y r' \over a r}e_{x y}' ={x (x - a) + y^2 \over y a}e_{x y}-{r r' \over a y}e_{x y}'.
p\bigl(D|x,a\bigr) = \Pr \biggl\{x_{t+1}\in D \biggl| x_t=x, a_t=a\biggr\}&= \Pr \biggl\{\theta_1 \min(a,x) \exp\{-\theta_2 \min(a,x) + v\}\in D\biggr\}&= \int_{D}g\biggl(\log(v) - \log(\theta_1 \min(a,x)) + \theta_2 \min(a,x)\biggr) \frac{1}{v}m(dv),
\beta L_j(\beta)-jL_j(\beta) =(\beta-j)L_j(\beta)=(\beta-j)\prod_{\substack{l=0 l\neq j}}^n\frac{\beta-l}{j-l}=\frac{\beta(\beta-1)\cdots(\beta-n)}{\prod_{\substack{l=0 l\neq j}}^n(j-l)}=0.
\dot{\ln{\frac{[ i ]}{[ i - 1 ]}}}+ ( \ln ( [ i ] [ i - 1 ] )^{\prime \prime}+ ( ( \ln{\frac{[ i ]}{[ i - 1 ]}}^{\prime})^{2}= 2 ( \bar{\alpha}_{i - 1 , i}+ \bar{\alpha}_{i , i - 1}- \bar{\alpha}_{i - 1}\bar{\alpha}_{i})
Z_{n^{( e )}}= \left( \begin{matrix}{{{\bf 1}}}&{{0}}\\{{{{\tilde{\bf c}}^{h}}}}&{{{\bf 1}}}\\ \end{matrix}\right) , \qquad \qquad{\tilde{\bf c}}^{h}= 4 0 n^{( e )}{n^{( e )}}^{t}
\lim_{N \to \infty}(2^{\frac{4}{3}}N^{\frac{1}{3}})^k R_{N}^{(k)}\Big(4N+2^{\frac{4}{3}}N^{\frac{1}{3}}u_1, \ldots, 4N+2^{\frac{4}{3}}N^{\frac{1}{3}}u_k\Big)= \det \big[K_{\mathrm{Airy}}(\pi;u_{i},u_{j})\big]_{i,j = 1}^{k}
X_d=2ig_L\sum_{\Xi\in\Delta_L}x_d(\Xi\cdot q, \xi)E_d(\Xi),\quad Y_d=ig_L\sum_{\Xi\in\Delta_L}y_d(\Xi\cdot q, \xi)E_d(\Xi),\quad E_d(\Xi)_{\Upsilon \Omega}=\delta_{\Upsilon-\Omega,2\Xi},
\sum_{n=2}^{\infty}I_{1}(n;t) \frac{x^{n}}{n!}= \sum_{n=1}^{\infty}tI_{1}(n;t) \frac{x^{n+1}}{(n+1)!}+ \sum_{n=0}^{\infty}I_{1}(n;t) \frac{x^{n+2}}{(n+1)n!}.
I_{1}=\ \left( \frac{(i)^{\alpha}+(-i)^{\alpha}}{4\cos\alpha\pi/2}\right) \int_{-\infty}^{+\infty}dq\frac{q^{\alpha}e^{i(\frac{n\pi x}{2a}+\frac{n\pi}{2})q}}{\ (q+1)(q-1)}\ , I_{2}=\ \left( \frac{(i)^{\alpha}+(-i)^{\alpha}}{4\cos\alpha\pi/2}\right) \int_{-\infty}^{+\infty}dq\frac{q^{\alpha}e^{i(\frac{n\pi x}{2a}-\frac{n\pi}{2})q}}{\ (q+1)(q-1)}\ ,
\|u\|_{6(\alpha-2)}\le C\|u\|_6^{1-\kappa}\|\Lambda^{s+1}u\|^\kappa = C\|u\|_6^{\frac{2(\alpha-2)s-\alpha+3}{2(\alpha-2)s}}\|\Lambda^{s+1}u\|^\frac{\alpha-3}{2(\alpha-2)s}.
\zeta_H'(2) =\sum_{n=2}^\infty\frac{H_n\log n}{n^2}\approx2.623865966, U =\int_0^1Z^2(t)dt\approx0.4785935, V =\sum_{n=1}^\infty\frac{\zeta(2n+1)}{n+1}E_{2n+2}\approx0.055645894.
\operatorname{Tail}_a(v ; x_{0}, r) :=\left(r^{sq}\underset{x\in B_{2r}(x_0)}{{\rm ess}\sup}\int_{\mathbb{R}^{N}\backslash B_{r}\left(x_{0}\right)}a(x,y) \frac{|v(y)|^{q-1}}{\left|y-x_{0}\right|^{N+s q}}\, d y\right)^{\frac{1}{q-1}}.
J_2(t) =n\sum_{l=1}^{n-1}(-\lambda_{l,n})^{-1}\|\{\Sigma_n(\mathbb U(t))-\Sigma_n(U(t))\}e_l\|^2 &\le C\left\|\mathbb U(t)-U(t)\right\|^2\le C\|\mathbb U(t)-\tilde U(t)\|^2+ C\|E(t)\|^2,
m^\mathrm{pair}(x,y) = -\sum_j{u_j(x)\overline{p_j(y)}}= \Big(k\circ(\delta-\overline{k}\circ k)^{-1}\Big)(x,y) n^\mathrm{pair}(x,y) = \sum_k{p_j(x)\overline{p_j(y)}}= \Big((k\circ\overline k)\circ(\delta- \overline{k}\circ k)^{-1}\Big)(x,y).
1 = \sigma_{-1}(N^\omega_{\partial M})(t,\xi')\sigma_1(\Lambda_{g,F}^\omega)(t,\xi')+\sigma_{-1}(N^\omega_{\partial M})(t,\xi')\sigma_0(\Lambda_{g,F}^\omega)(t,\xi'), &+\sigma_{-2}(N^\omega_{\partial M})(t,\xi')\sigma_1(\Lambda_{g,F}^\omega)(t,\xi') +\nabla_{\xi'}\sigma_{-1}(N^\omega_{\partial M}) \cdot D_{t'}\sigma_1(\Lambda_{g,F}^\omega) + S^{-2}_{1,0}.
\left(f+\psi_{\varepsilon}\circ L\right)^{*}_\Phi\left(0\right)+f\left(x_\varepsilon\right)+\psi_{\varepsilon}\left(Lx_\varepsilon\right)=f^{*}_\Phi \left(-\psi_\varepsilon \circ L \right)+f\left(x_\varepsilon\right)+ \psi_{\varepsilon}(L x_\varepsilon) \leq\varepsilon
\sum_{j=0}^{n}Y_{j}\left( \lambda \right) Y_{n-j}\left( \lambda \right) =\frac{\left( -1\right)^{n}\left( n+1\right) !}{2^{n-2}\left( \lambda-1\right)^{2}}\left( \frac{\lambda^{2}}{\lambda -1}\right)^{n}\sum_{j=0}^{n}\frac{2^{j}}{j+1}.
G(\epsilon)=\exp\left[\frac{V}{(2\pi)^{(n-1)}}\sum_{\sigma}\int d^{n-1}p\;\epsilon(p)[d_p^{(\sigma)}\bar{d}_{\tilde p}^{(-\sigma)}-d_p^{(\sigma)\,\dagger}\bar{d}_{\tilde p}^{(-\sigma)\,\dagger}]\right]\,{,}
\sum_{t=T_{1}+1}^{T}\mathbb{E}[f(x_{t})-f^*]\leq \sum_{t=T_{1}+1}^{T}\Big[\frac{r_{t}-r_{t+1}}{2\eta_t}- \frac{\alpha}{4}r_t\Big]+\sum_{t=T_{1}+1}^{T}\Big[2(\kappa_{\beta}L)^2\frac{d}{\alpha}h_{t}^{2(\beta-1)}+\frac{3\eta_t}{4}\kappa\left( \frac{\sigma^2}{h_{t}^2}+ \frac{3{\bar L}^2}{2}h_{t}^2 \right)d \Big],
A \cdot \begin{pmatrix}a_1 a_2 \vdots a_n \end{pmatrix}= 0, A = \begin{pmatrix}1 1^2 \cdots 1^n 2 2^2 \cdots 2^n \vdots \vdots \ddots \vdots n n^2 \cdots n^n \end{pmatrix}
\eta(\tau_n^q\tau_j, \tau_i)=\eta(\tau_j, \tau_n \tau_{i+1})=\eta(\tau_j, \tau_n) \eta(\eta(\tau_n, \tau_j), \tau_{i+1})=\tau_n^q \tau_1 \eta(\tau_n^{q-1+D(j)}\tau_{j+1},\tau_{i+1})
\psi_{\alpha}(x)=\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{\sqrt{2E_{p}}}\sum_{s}(a_{p}^{s}u_{\alpha}^{s}(p)e^{-ip\cdot x}+b_{p}^{s\dagger}v_{\alpha}^{s}(p)e^{ip\cdot x}).
\delta = \sum_{Y\in \mathcal{I}_X^G}l_{\delta}(u_Y,0) \overline{Y\times \mathbb{C}^*}+ \sum_{D\in \mathcal{D}_X}l_{\delta}(\rho(D),0) \overline{D\times \mathbb{C}^*}& + \sum_{D\in \mathcal{D}\setminus \mathcal{D}_X}an_D \overline{D\times \mathbb{C}^*}+ l_{\delta}(0,-1)X_{\infty}+ l_{\delta}(\lambda,m) X_0.
R(\phi) = \sum_{n=0}^{\infty}r^{(n)}_{m_{1}\ldots m_{n}}\phi_{m_{1}}\ldots \phi_{m_{n}}, \;\;\;\;\;\; \tilde{R}(\tilde{\phi}) = \sum_{n=0}^{\infty}r^{(n)}_{m_{1}\ldots m_{n}}\tilde{\phi}_{m_{n}}\ldots \tilde{\phi}_{m_{1}}
\begin{array}{r l}{{E_{8}^{+}=}}&{{\{\pm e_{a}\}}}\\{{\cup}}&{{\{( \pm e_{a}\pm e_{b}\pm e_{c}\pm e_{d}) / 2 : a , b , c , d \mathrm{~ d i s t i n c t}, \; e_{a}( e_{b}( e_{c}e_{d}) ) = \pm 1 \},}}\\{{}}&{{a , b , c , d \in \{0 , . . . , 7 \}.}}\\ \end{array}
\left(\int u^{2}(x) d\nu(x)\right)^{1/2}= \left(\int_{y}(u+v)^{2}(y) \int_{x}u^{2}(x) d\mu(x,y)\right)^{1/2}&= \left( \int_{y}(u+v)^{2}(y) d\nu_{u}(y) \right)^{1/2}&\leq \left( \int_{y}u^{2}(y) d\nu_{u}(y) \right)^{1/2}+ \left( \int_{y}v^{2}(y) d\nu_{u}(y) \right)^{1/2}&= B[u^{2}]^{1/2}+b[u^{2},v^{2}]^{1/2},
\Re\sum_{j=1}^r\frac{y^j(e^{itj}-1)}{j}\leq \frac{1}{r}\Re \sum_{j=1}^ry^j(e^{itj}-1) &= \frac{y^{r+1}}{r(y-1)}\left(\Re \frac{e^{it(r+1)}(y-1)}{ye^{it}-1}-1\right)+\frac{y}{r(y-1)}\left(1-\Re \frac{e^{it}(y-1)}{ye^{it}-1}\right) &\leq\frac{y^{r+1}}{r(y-1)}\left(\frac{y-1}{|ye^{it}-1|}-1\right)+\frac{2y}{r(y-1)}.
R_{N}^{( \pm )}( \nu , k ) = \int_{1}^{\infty}d \alpha ( \sum_{\ell = 1}^{2 N + 3}\frac{\Phi_{\ell}^{( N + 1 )}( \alpha )}{( \alpha \pm i k )^{\ell})}[ \alpha^{\nu / 2}+ \alpha^{- \nu / 2}] ,
\textbf{T}_\alpha^a I_\alpha^a f(t)= T_\beta^a (\frac{d^n}{dt^n}I_\alpha^a f(t))=T_\beta^a (\frac{d^n}{dt^n}I_{n+1}^a ((t-a)^{\beta-1}f(t) ))=T_\beta^a ( I_{1}^a ( (t-a)^{\beta-1}f(t)))
\textbf{y}(t)= e^{A \frac{(t-a)^\alpha}{\alpha}}\textbf{c}+ \int_a^t e^{A \frac{(t-a)^\alpha}{\alpha}}e^{-A \frac{(s-a)^\alpha}{\alpha}}\textbf{f}(s) (s-a)^{1-\alpha}ds,
\frac{\partial}{\partial t}\Big( \int_M |u|^p d \mu \Big) = 0 = \frac{\partial}{\partial t}\Big( \int_M |u|^{p-1}u d \mu \Big) \displaystyle = \int_M(p-1) |u|^{p-2}u \frac{\partial u}{\partial t}d\mu + \int_M |u|^{p-1}\frac{\partial}{\partial t}(u d\mu).
M_{\Omega_1}= \frac{1}{\alpha}(D_{A\Omega_1}- \beta L_{A\Omega_1}), N_{\Omega_1}= \frac{1}{\alpha}[(1-\alpha)D_{A\Omega_1}+(\alpha-\beta) L_{A\Omega_1}+\alpha U_{A\Omega_1}]; M_{\Omega_2}= D_{A\Omega_1}- U_{A\Omega_1}, N_{\Omega_2}= L_{A\Omega_1},
({\bf x}t \vert{\bf x}^{\prime}t^{\prime}) = \frac{1}{\sqrt{2 \pi i \epsilon \hbar / M}^{D}}\prod_{n = 1}^{N}\left[ \int_{- \infty}^{\infty}d x_{n}\right] \prod_{n = 1}^{N + 1}K_{0}^{\epsilon}( \Delta{\bf x}_{n}) ,
{\frac{1}{[ q^{+}]_{M L}}}\equiv{\frac{1}{q^{+}+ i \epsilon \ \mathrm{s i g n}( q^{-})}}={\frac{1}{[ q^{+}]_{C P V}}}- i \pi \ \mathrm{s i g n}( q^{-}) \delta ( q^{+}) ,
\xi_{i}= 0 \textnormal{if}\chi_{i}^{2}(g) \neq \frac{\psi (g ,\, g_{i}^{2})}{\psi (g_{i}^{2}, g)}, \alpha_{ij}= 0 \textnormal{if}\chi_{i}\chi_{j}(g) \neq \frac{\psi (g,\, g_{i}g_{j})}{\psi (g_{i}g_{j},\, g)},
\frac{\eta^+\cdot\sigma}{|\eta^+|}= \frac{|\eta^+|}{|\eta|}\frac{\eta\cdot\sigma}{|\eta|}= 2\frac{|\eta^+|^2}{|\eta|^2}- 1 = 2 \left(\frac{\eta^+\cdot\sigma}{|\eta^+|}\right)^2 - 1.
\nu_{N_1+N_2}^{\sigma}(s) (dx) : = \frac{1}{Z}\exp \left( \sigma \sum_{k=1}^{N_1+N_2}w_k - H_{N_1+N_2}(w) + s \sum_{(i, j) \in I_{N_1, N_2}}M_{ij}u_i v_j \right).
\hat{\tilde{g}}(y,\xi, \lambda) = \int_{0}^{\infty}e^{- \lambda t}\int_{-\infty}^{+\infty}e^{i \xi x}g(x,y,t) \, \mathrm{d}x \,\mathrm{d}t &= \int_{0}^{\infty}e^{- \lambda t}e^{i \xi y \,- \,\xi^{2}t}\,\mathrm{d}t &= \frac{e^{i \xi y}}{\lambda+ \xi^{2}}.
\lim_{N \to \infty}\iint_{(C, \infty)^2}dx dr \left\lvert G_N(x, r) - G_{\infty, \gamma}(x, r) \right\rvert^2 ={}0, \lim_{N \to \infty}\iint_{(C, \infty)^2}dr dy \left\lvert H_N(r, y) - H_{\infty, \gamma}(r, y) \right\rvert^2 ={}0.
\mbox{Eq.~(\ref{hinagata})}\simeq A B C + \frac{i}{2}\left[ \frac{\partial A}{\partial P^\mu}\frac{\partial B}{\partial X_\mu}C - A \frac{\partial B}{\partial X_\mu}\frac{\partial C}{\partial P^\mu}- \frac{\partial A}{\partial X_\mu}\frac{\partial B C}{\partial P^\mu}+ \frac{\partial A B}{\partial P^\mu}\frac{\partial C}{\partial X_\mu}\right] ,
\left(\dfrac{1}{(\sin \sqrt{\kappa}s)^{\frac{m-1}{2}}}\int_{M\cap B_s}S_1dM\right)^{1/p}\leq\left(\dfrac{1}{(\sin \sqrt{\kappa}t)^{\frac{m-1}{2}}}\int_{M\cap B_t}S_1dM\right)^{1/p}&\qquad+ \frac{\Lambda(m-1)}{c^{1-1/p}(m-1-2p)}\int_s^t\frac{1}{(\sin \sqrt{\kappa}r)^{\frac{m-1}{2p}}}dr.
\tilde{\chi}^{\mu_{1}\ldots \mu_{p - 1}}= \partial_{\mu}A^{\mu \mu_{1}\ldots \mu_{p - 1}}+ \frac 1{p - 1}\partial^{\left[ \mu_{1}\right.}B^{( 1 ) \left. \mu_{2}\ldots \mu_{p - 1}\right]},
H^{p,q}(C_1 \times C_2 \times C_3)\simeq\bigoplus_{\substack{s_1+s_2+s_3=p t_1 + t_2+t_3 =q}}H^{s_1,t_1}(C_1) \otimes H^{s_2,t_2}(C_2) \otimes H^{s_3,t_3}(C_3),
\frac{1}{2}\sum_{m , n = - \infty}^{\infty}: \tilde{\alpha}_{- ( m + n )}^{( A )}\tilde{\alpha}_{n ( A )}: ( c_{m}+ \tilde{c}_{- m}) | B ; p , 0 \rangle = 0 \, \, ; \, \, A \in \{0 , . . . , 2 5 \}
\lambda^2\frac{1}{2^{15}3^{3}\pi^{6}}\left[ \sum_{N=1}^{\infty}[C_N^{\frac{3}{2}}(cos{\theta_4^0})]^2\frac{(N+1)(N+2)}{N(N+3)}\right] (RK^2)\frac{1}{\epsilon}\int_C d^4{\eta}+ F.T.
\Gamma_+\, C^{-1}\, \tilde\Gamma_+ = (U_3 + \sum_1^2 A^{(r)}U_rA^{(r)T})\, C^{-1}\, (U_3^{-1}+ \sum_1^2 A^{(s)}U_s^{-1}A^{(s)T})
V_0(\lambda) ={(-1)^{\eta-1}\over (4\pi)^\eta \eta! z_-^{D-1}}\left[ \left\{\ln(\ell\mu z_-)+{1\over 2}\sum_{n=1}^{\eta}n^{-1}\right\}\hat\zeta(1-D) +\hat\zeta'(1-D)\right].
\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{A_0(x,y)}{|x-y|^{n+2s}}(\ell(x)-\ell(y))(\varphi(x)-\varphi(y))dydx & = 2 \int_{\mathbb{R}^n}\left (\int_{\mathbb{R}^n}\frac{A_0(x,y)}{|x-y|^{n+2s}}(\ell(x)-\ell(y)) dy \right ) \varphi(x)dx =0.
(0,1-a,0,0)\otimes(0,1-a,0,0) = (0,1,0,0)\otimes(0,1,0,0)-(0,a,0,0)\otimes(0,1,0,0) - (0,1,0,0)\otimes(0,a,0,0)+(0,a,0,0)\otimes(0,a,0,0).
=\frac{1}{2}(m_{1}v_{0}^{2}-2pv_{0}+\frac{p^{2}}{m_{1}}+m_{2}v_{0}^{2}+2pv_{0}+\frac{p^{2}}{m_{2}}-m_{1}v_{0}^{2}-m_{2}v_{0}^{2})
B_{s,l}[i][j] =\frac{\beta_s[i]\beta_u[j]}{F[i][j](\beta_s[i]+\theta_{s,l}[i])\ln2}> 0, B_{u,l}[i][j] =\frac{\beta_s[i]\beta_u[j]}{F[i][j](\beta_u[j]+\theta_{u,l}[j])\ln2}> 0,
\begin{aligned}s_{ik}= G_{ik}(z,\delta) := \sum_{t=1}^{l_i}\Bigl( m^t_{i0}(\delta_{it-1}- \delta_{it}) + \sum_{j=1}^k(m^t_{ij}- m^t_{ij-1})(z_{ij}- \delta_{it})\Bigr), \end{aligned}
\mathbf a = (\mathbf a_1 (1), \mathbf a_1 (2), \mathbf a_1 (3), \mathbf a_2 (1)) &= (0, \langle \lambda_1, \alpha_2^{\vee}\rangle + \langle \lambda_2, \alpha_2^{\vee}\rangle,\langle \lambda_1, \alpha_1^{\vee}\rangle + \langle \lambda_2, \alpha_1^{\vee}\rangle,\langle \lambda_2, \alpha_{3}^{\vee}\rangle).
\begin{array}{ll}p^{m}(p^m-1)(2p^{m+d}-2p^m+2p^d-p^{2d}-1) = (p^d-1)p^{m}((p^{\frac{d}{2}}+1)^2(n_{1,1,1}+n_{-1,-1,1})+(p^{\frac{d}{2}}-1)^2(n_{1,-1,1}+n_{-1,1,1}) &+(p^d+1)^3(n_{1,1,2}+n_{-1,-1,2})+(p^d+1)(p^d-1)^2(n_{1,-1,2}+n_{-1,1,2})).\end{array}
\frac{1}{p}+\frac{1}{1+\epsilon_{0}}=\frac{1}{\tilde{p}}\mbox{and}\frac{1}{n+1}+\frac{1}{p}\leq \frac{1}{\tilde{p}}<\frac{1}{p}+\min\left\{\frac{1}{p},\frac{1}{n}\right\}.
\left| \bar{K}(x,y) - \bar{K}(x',y) \right| = \left| K(0,y-x) - K(x',y) \right| \leq \left|K(0,y-x) - K(0,y)\right| + \left|K(0,y) - K(x',y)\right| \leq \kappa |x|^\delta|x-y|^{-1+s-\delta}+ \kappa |x'|^\delta |x'-y|^{-1+s-\delta}\leq 5 \kappa |x-x'|^\delta|x-y|^{-1+s-\delta}.